Abstract group 9600.cb: $D_4\times D_5\times S_5$

• Label: 9600.cb
• Order: \( 2^{7} \cdot 3 \cdot 5^{2} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3 \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $14$
• Trans deg.: $100$
• Rank: $4$

Group information

Description:	$D_4\times D_5\times S_5$
Order:	\(9600\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{2} \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_5^4:C_2.C_2^4$, of order \(76800\)\(\medspace = 2^{10} \cdot 3 \cdot 5^{2} \)
Composition factors:	$C_2$ x 5, $C_5$, $A_5$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30	60
Elements	1	935	20	1752	124	1420	1940	480	80	1648	880	320	9600
Conjugacy classes	1	23	1	16	5	15	35	4	2	20	14	4	140
Divisions	1	23	1	16	3	15	21	4	1	11	7	2	105
Autjugacy classes	1	13	1	12	3	8	13	4	1	9	4	2	71

Dimension	1	2	4	5	6	8	10	12	16	20	24	32	40	48
Irr. complex chars.	16	20	20	16	8	20	20	10	4	4	2	0	0	0	140
Irr. rational chars.	16	4	24	16	8	6	4	2	8	8	4	2	2	1	105

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$100$
Rank:	$4$
Inequivalent generating quadruples:	$33294431520$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	16	32
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $14$ $\langle(2,4)(3,5), (1,14,11,12,13)(2,3,5,4), (2,3)(4,5), (1,11)(3,4)(6,10,8)(7,9)(12,13), (1,14)(2,5)(3,4)(11,13), (2,5), (1,14)(2,5)(3,4)(9,10)(11,13)\rangle$
Direct product:	$D_4$ $\, \times\, $ $D_5$ $\, \times\, $ $S_5$
Semidirect product:	$(C_4\times S_5)$ $\,\rtimes\,$ $D_{10}$	$(C_4:S_5)$ $\,\rtimes\,$ $D_{10}$	$D_{20}$ $\,\rtimes\,$ $(C_2\times S_5)$	$(D_4\times A_5)$ $\,\rtimes\,$ $D_{10}$	all 32
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{10}$ . $(C_2^2\times S_5)$	$(C_{10}\times S_5)$ . $C_2^3$	$(C_{10}:S_5)$ . $C_2^3$	$C_{10}$ . $(C_2^3\times S_5)$	all 10

Elements of the group are displayed as permutations of degree 14.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 198744 subgroups in 4405 conjugacy classes, 122 normal (50 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_{10}.(C_2\times S_5)$
Commutator:	$G' \simeq$ $C_{10}\times A_5$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_{10}.(C_2\times S_5)$
Fitting:	$\operatorname{Fit} \simeq$ $C_5\times D_4$	$G/\operatorname{Fit} \simeq$ $C_2\times S_5$
Radical:	$R \simeq$ $D_4\times D_5$	$G/R \simeq$ $S_5$
Socle:	$\operatorname{soc} \simeq$ $C_{10}\times A_5$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 9600: $D_4\times D_5\times S_5$

Order 4800: $C_5:D_4.S_5$ x 2, $C_5:D_4.S_5$ x 2, $C_2^2\times D_5.S_5$ x 2, $D_5.C_2^2:S_5$ x 2, $D_{20}.S_5$, $D_{20}.S_5$, $(C_5\times D_4).S_5$, $(C_4\times D_5).S_5$, $(D_4\times D_5).A_5$, $(C_5\times D_4).S_5$, $(C_4\times D_5).S_5$

Order 2400: $C_2\times D_5.S_5$ x 15, $D_{10}.S_5$ x 2, $C_5.C_2^2:S_5$ x 2, $C_5.C_2^2:S_5$ x 2, $C_2^2\times C_5.S_5$ x 2, $C_5:C_4.S_5$ x 2, $(C_2^2\times D_5).A_5$ x 2, $C_5:D_4.A_5$ x 2, $D_{10}.S_5$ x 2, $(C_2\times C_{10}).S_5$ x 2, $C_5:C_4.S_5$, $C_{20}.S_5$, $D_{20}.A_5$, $(C_5\times D_4).A_5$, $C_{20}.S_5$, $C_{20}.S_5$, $C_{20}.S_5$, $C_5:C_4.S_5$, $(C_4\times D_5).A_5$, $D_{10}.S_5$

Order 1920: $C_2\times D_4\times S_5$, $\GL(2,\mathbb{Z}/4):D_{10}$

Order 1600: $D_{10}^2.C_2^2$

Order 1200: $D_5\times S_5$ x 16, $D_{10}\times A_5$ x 7, $C_{10}\times S_5$ x 7, $C_{10}:S_5$ x 7, $C_2\times C_{10}\times A_5$ x 2, $C_{20}\times A_5$, $C_5:C_4\times A_5$, $C_{10}.S_5$, $A_5:C_{20}$

Order 960: $D_4\times S_5$ x 8, $C_2\times D_{10}\times S_4$ x 2, $C_2^3\times S_5$ x 2, $\GL(2,\mathbb{Z}/4):D_5$ x 2, $C_5:D_4\times S_4$ x 2, $D_5\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^3:S_5$ x 2, $C_{60}:C_2^4$, $D_4\times D_5\times A_4$, $C_2^4:D_{30}$, $\GL(2,\mathbb{Z}/4):C_{10}$, $D_{20}:S_4$, $S_4\times D_{20}$, $D_5\times C_4:S_4$, $C_4\times D_5\times S_4$, $C_2\times D_4\times A_5$, $C_2\times C_4:S_5$, $C_2\times C_4\times S_5$

Order 800: $D_{10}^2.C_2$ x 2, $D_{10}:(C_4\times D_5)$ x 2, $D_{10}^2.C_2$ x 2, $C_5:D_4\times F_5$ x 2, $D_{20}:F_5$, $D_5^2.D_4$, $F_5\times D_{20}$, $C_4\times D_5\times F_5$, $D_4\times D_5^2$, $D_4\times C_5:F_5$, $D_{20}:C_{20}$

Order 640: $D_5\times D_4^2$

Order 600: $D_5\times A_5$ x 4, $C_5:S_5$ x 4, $C_5\times S_5$ x 4, $C_{10}\times A_5$ x 3

Order 480: $C_2^2\times S_5$ x 19, $D_{12}:D_{10}$ x 16, $D_{10}\times S_4$ x 15, $C_2^2:S_5$ x 8, $D_4\times A_5$ x 4, $C_4:S_5$ x 4, $C_4\times S_5$ x 4, $D_{10}:S_4$ x 2, $D_{10}:S_4$ x 2, $(C_2\times C_{10}):D_{12}$ x 2, $C_{15}:C_2^5$ x 2, $C_2\times A_4\times D_{10}$ x 2, $C_2^3:D_{30}$ x 2, $C_2\times C_{10}\times S_4$ x 2, $C_2^3\times A_5$ x 2, $D_{30}:D_4$ x 2, $D_{30}:C_2^3$ x 2, $D_{30}:C_2^3$ x 2, $A_4\times C_5:D_4$ x 2, $C_5:\GL(2,\mathbb{Z}/4)$ x 2, $C_5\times \GL(2,\mathbb{Z}/4)$ x 2, $D_{10}.S_4$, $C_5:(C_4\times S_4)$, $C_5:C_4\times S_4$, $C_2\times C_4\times A_5$, $C_2^2.S_5$, $D_4\times D_{30}$, $C_{60}:C_2^3$, $C_{60}:C_2^3$, $C_2^4:C_{30}$, $D_{12}:D_{10}$, $D_6\times D_{20}$, $D_{10}\times D_{12}$, $C_{60}:C_2^3$, $A_4\times D_{20}$, $C_4\times D_5\times A_4$, $C_{20}:S_4$, $C_{20}:S_4$, $C_{20}:S_4$, $C_{20}\times S_4$

Order 400: $D_{10}\times F_5$ x 16, $D_{10}:D_{10}$ x 4, $D_{10}:F_5$ x 4, $D_{10}^2$ x 2, $C_{10}^2:C_4$ x 2, $D_{10}:C_{20}$ x 2, $C_{20}:D_{10}$ x 2, $D_{10}:D_{10}$ x 2, $D_5\times D_{20}$ x 2, $C_{10}^2:C_4$ x 2, $D_{10}:C_{20}$ x 2, $D_{10}.D_{10}$ x 2, $C_5^2:C_4^2$ x 2, $C_{20}:D_{10}$, $C_{20}:D_{10}$, $C_4\times D_5^2$, $C_{20}:F_5$, $C_{20}:F_5$, $C_{20}:C_{20}$, $C_{20}\times F_5$

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$

Order 320: $C_{20}:C_2^4$ x 4, $C_2^4:D_{10}$ x 4, $D_{20}:D_4$ x 4, $C_4:D_4\times D_5$ x 4, $C_2^4:D_{10}$ x 4, $C_2^4:D_{10}$ x 4, $D_4\times D_{20}$ x 2, $C_4^2:D_{10}$ x 2, $D_{10}.C_2^4$, $C_5\times D_4^2$, $C_4^2:D_{10}$, $C_4^2:D_{10}$

Order 300: $C_5\times A_5$

Order 240: $D_6\times D_{10}$ x 39, $C_2\times S_5$ x 30, $D_5\times S_4$ x 16, $D_6:D_{10}$ x 16, $D_6:D_{10}$ x 16, $D_6:D_{10}$ x 16, $C_2^2\times A_5$ x 9, $D_4\times D_{15}$ x 8, $C_{20}:D_6$ x 8, $C_{12}:D_{10}$ x 8, $C_{12}:D_{10}$ x 8, $S_3\times D_{20}$ x 8, $D_5\times D_{12}$ x 8, $C_{12}:D_{10}$ x 8, $A_4\times D_{10}$ x 7, $C_{10}:S_4$ x 7, $C_{10}\times S_4$ x 7, $C_4\times A_5$ x 2, $A_5:C_4$ x 2, $C_2^2\times D_{30}$ x 2, $C_{15}:C_2^4$ x 2, $C_{15}:C_2^4$ x 2, $C_2^3:C_{30}$ x 2, $C_{30}:D_4$ x 2, $C_{30}:D_4$ x 2, $C_{30}:D_4$ x 2, $C_{10}:D_{12}$ x 2, $C_6:D_{20}$ x 2, $C_{30}:D_4$ x 2, $D_4\times C_{30}$, $C_2\times D_{60}$, $C_4\times D_{30}$, $C_{10}\times D_{12}$, $D_6\times C_{20}$, $C_6\times D_{20}$, $C_{12}\times D_{10}$, $A_4\times C_{20}$, $D_{30}:C_4$, $D_6.D_{10}$, $D_{10}.D_6$, $A_4\times C_5:C_4$, $C_{10}.S_4$, $A_4:C_{20}$

Order 200: $D_5\times F_5$ x 18, $D_5\times D_{10}$ x 15, $C_{10}:F_5$ x 8, $C_{10}\times F_5$ x 8, $C_{10}\times D_{10}$ x 4, $C_{10}\wr C_2$ x 4, $C_5:D_{20}$ x 4, $C_{10}:D_{10}$ x 2, $C_{10}^2:C_2$ x 2, $C_5\times D_{20}$ x 2, $D_5\times C_{20}$ x 2, $D_{10}:D_5$ x 2, $C_{10}.D_{10}$ x 2, $D_4\times C_5^2$, $C_5:D_{20}$, $C_{20}:D_5$, $C_{10}.D_{10}$

Order 192: $D_4\times S_4$ x 8, $C_2^3\times S_4$ x 2, $C_2\times \GL(2,\mathbb{Z}/4)$ x 2, $C_{12}:C_2^4$, $C_2^5:C_6$, $C_2^3:D_{12}$, $C_2^4.D_6$

Order 160: $D_4\times D_{10}$ x 57, $D_{10}:D_4$ x 12, $C_2^2:D_{20}$ x 12, $C_2^3:D_{10}$ x 8, $D_4\times F_5$ x 8, $C_2^3.D_{10}$ x 8, $D_{10}:D_4$ x 8, $D_{10}.D_4$ x 8, $C_{20}:D_4$ x 6, $C_2^3\times D_{10}$ x 4, $C_{20}:C_2^3$ x 4, $C_2^2\times D_{20}$ x 4, $C_{20}:C_2^3$ x 4, $C_{20}:D_4$ x 4, $C_2^4:C_{10}$ x 4, $C_2^4:D_5$ x 4, $C_{20}:D_4$ x 4, $C_{20}:D_4$ x 4, $C_{20}:D_4$ x 4, $C_4:D_{20}$ x 4, $C_2^3.D_{10}$ x 4, $C_4\times D_{20}$ x 2, $C_2^3\times F_5$ x 2, $C_2^3:F_5$ x 2, $D_4\times C_{20}$ x 2, $C_2^3.D_{10}$ x 2, $D_{20}:C_4$ x 2, $D_{10}.D_4$ x 2, $C_4:D_{20}$, $D_5\times C_4^2$, $D_{10}.D_4$, $C_{10}:C_4^2$, $C_{20}:D_4$

Order 128: $C_2\times D_4^2$

Order 120: $S_3\times D_{10}$ x 124, $C_2\times D_{30}$ x 19, $C_{10}\times D_6$ x 19, $C_6\times D_{10}$ x 19, $S_5$ x 8, $C_{15}:D_4$ x 8, $C_{15}:D_4$ x 8, $C_{15}:D_4$ x 8, $C_5:D_{12}$ x 8, $C_3:D_{20}$ x 8, $C_{15}:D_4$ x 8, $C_2\times A_5$ x 7, $C_6.D_{10}$ x 4, $C_6.D_{10}$ x 4, $D_5\times A_4$ x 4, $C_5:S_4$ x 4, $C_5\times S_4$ x 4, $D_4\times C_{15}$ x 4, $D_{60}$ x 4, $C_4\times D_{15}$ x 4, $C_5\times D_{12}$ x 4, $S_3\times C_{20}$ x 4, $C_3\times D_{20}$ x 4, $D_5\times C_{12}$ x 4, $D_{15}:C_4$ x 4, $C_{10}\times A_4$ x 3, $C_2^2\times C_{30}$ x 2, $C_2\times C_{60}$, $C_{30}:C_4$, $C_6:C_{20}$, $C_{10}:C_{12}$

Order 100: $D_5^2$ x 16, $C_5\times D_{10}$ x 14, $C_5:D_{10}$ x 7, $C_5\times F_5$ x 5, $C_5:F_5$ x 5, $C_5:C_{20}$ x 2, $C_{10}^2$ x 2, $C_5\times C_{20}$, $C_5^2:C_4$

Order 96: $D_4\times D_6$ x 24, $C_2^2\times S_4$ x 19, $\GL(2,\mathbb{Z}/4)$ x 8, $D_4\times A_4$ x 4, $C_4:S_4$ x 4, $C_4\times S_4$ x 4, $C_2^3\times D_6$ x 2, $C_2^3\times A_4$ x 2, $C_2^3:D_6$ x 2, $C_{12}:C_2^3$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$, $C_2^3:C_{12}$, $C_2^2.S_4$

Order 80: $D_4\times D_5$ x 120, $C_{10}:D_4$ x 58, $C_2^2\times D_{10}$ x 50, $D_4\times C_{10}$ x 29, $C_2\times D_{20}$ x 29, $C_4\times D_{10}$ x 28, $C_2^2\times F_5$ x 20, $D_{10}:C_4$ x 16, $D_{10}:C_4$ x 8, $C_2^2:C_{20}$ x 8, $C_{10}.D_4$ x 8, $C_2^3\times C_{10}$ x 4, $C_2^2\times C_{20}$ x 4, $C_2^2.D_{10}$ x 4, $C_{20}:C_4$ x 4, $C_4\times F_5$ x 4, $C_{10}.D_4$ x 4, $C_{20}:C_4$ x 3, $C_4:C_{20}$ x 2, $C_{20}:C_4$ x 2, $C_4\times C_{20}$

Order 64: $D_4^2$ x 16, $D_4\times C_2^3$ x 4, $C_2^3:D_4$ x 4, $C_2^3:D_4$ x 4, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$

Order 60: $S_3\times D_5$ x 64, $D_{30}$ x 30, $S_3\times C_{10}$ x 30, $C_3\times D_{10}$ x 30, $C_2\times C_{30}$ x 9, $C_{60}$ x 2, $C_{15}:C_4$ x 2, $C_5:C_{12}$ x 2, $C_3:C_{20}$ x 2, $C_5\times A_4$, $A_5$

Order 50: $C_5\times D_5$ x 8, $C_5:D_5$ x 4, $C_5\times C_{10}$ x 3

Order 48: $S_3\times D_4$ x 64, $C_2^2\times D_6$ x 43, $C_2\times S_4$ x 30, $C_6:D_4$ x 24, $C_6\times D_4$ x 12, $C_2\times D_{12}$ x 12, $C_4\times D_6$ x 12, $C_2^2\times A_4$ x 9, $C_2^3\times C_6$ x 2, $C_4\times A_4$ x 2, $A_4:C_4$ x 2, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 40: $C_2\times D_{10}$ x 158, $C_5:D_4$ x 68, $D_{20}$ x 34, $C_4\times D_5$ x 34, $C_2\times F_5$ x 34, $C_5\times D_4$ x 34, $C_2^2\times C_{10}$ x 24, $C_2\times C_{20}$ x 14, $C_{10}:C_4$ x 14

Order 32: $C_2^2\times D_4$ x 72, $C_4:D_4$ x 32, $C_2^2\wr C_2$ x 32, $C_4\times D_4$ x 16, $C_4:D_4$ x 8, $C_2^3:C_4$ x 8, $C_2^5$ x 4, $C_2^3\times C_4$ x 4, $C_2^2.D_4$ x 2, $C_2\times C_4^2$

Order 30: $D_{15}$ x 8, $C_3\times D_5$ x 8, $C_5\times S_3$ x 8, $C_{30}$ x 7

Order 25: $C_5^2$

Order 24: $C_2\times D_6$ x 162, $C_3:D_4$ x 32, $C_2^2\times C_6$ x 21, $D_{12}$ x 16, $C_4\times S_3$ x 16, $C_3\times D_4$ x 16, $S_4$ x 8, $C_2\times A_4$ x 7, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6

Order 20: $D_{10}$ x 128, $C_2\times C_{10}$ x 44, $C_{20}$ x 11, $C_5:C_4$ x 11, $F_5$ x 10

Order 16: $C_2\times D_4$ x 223, $C_2^4$ x 52, $C_2^2\times C_4$ x 35, $C_2^2:C_4$ x 32, $C_4:C_4$ x 8, $C_4^2$ x 4

Order 15: $C_{15}$

Order 12: $D_6$ x 124, $C_2\times C_6$ x 39, $C_{12}$ x 4, $C_3:C_4$ x 4, $A_4$

Order 10: $D_5$ x 24, $C_{10}$ x 21

Order 8: $C_2^3$ x 159, $D_4$ x 116, $C_2\times C_4$ x 55

Order 6: $S_3$ x 16, $C_6$ x 15

Order 5: $C_5$ x 3

Order 4: $C_2^2$ x 124, $C_4$ x 16

Order 3: $C_3$

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 9600: $D_4\times D_5\times S_5$

Order 4800: $D_{20}.S_5$, $D_{20}.S_5$, $C_5:D_4.S_5$, $C_5:D_4.S_5$, $(C_5\times D_4).S_5$, $C_2^2\times D_5.S_5$, $D_5.C_2^2:S_5$, $(C_4\times D_5).S_5$, $(D_4\times D_5).A_5$, $(C_5\times D_4).S_5$, $(C_4\times D_5).S_5$

Order 2400: $C_2\times D_5.S_5$ x 5, $D_{10}.S_5$, $C_5.C_2^2:S_5$, $C_5:C_4.S_5$, $C_{20}.S_5$, $C_5.C_2^2:S_5$, $D_{20}.A_5$, $(C_5\times D_4).A_5$, $C_2^2\times C_5.S_5$, $C_{20}.S_5$, $C_5:C_4.S_5$, $(C_2^2\times D_5).A_5$, $C_{20}.S_5$, $C_5:D_4.A_5$, $C_{20}.S_5$, $C_5:C_4.S_5$, $(C_4\times D_5).A_5$, $D_{10}.S_5$, $D_{10}.S_5$, $(C_2\times C_{10}).S_5$

Order 1920: $C_2\times D_4\times S_5$, $\GL(2,\mathbb{Z}/4):D_{10}$

Order 1600: $D_{10}^2.C_2^2$

Order 1200: $D_5\times S_5$ x 4, $D_{10}\times A_5$ x 3, $C_{10}\times S_5$ x 3, $C_{10}:S_5$ x 3, $C_2\times C_{10}\times A_5$, $C_{20}\times A_5$, $C_5:C_4\times A_5$, $C_{10}.S_5$, $A_5:C_{20}$

Order 960: $D_4\times S_5$ x 6, $C_2\times D_{10}\times S_4$, $C_2^3\times S_5$, $C_{60}:C_2^4$, $D_4\times D_5\times A_4$, $C_2^4:D_{30}$, $\GL(2,\mathbb{Z}/4):C_{10}$, $\GL(2,\mathbb{Z}/4):D_5$, $C_5:D_4\times S_4$, $D_5\times \GL(2,\mathbb{Z}/4)$, $D_{20}:S_4$, $S_4\times D_{20}$, $D_5\times C_4:S_4$, $C_4\times D_5\times S_4$, $C_2\times D_4\times A_5$, $C_2^3:S_5$, $C_2\times C_4:S_5$, $C_2\times C_4\times S_5$

Order 800: $D_{20}:F_5$, $D_5^2.D_4$, $F_5\times D_{20}$, $C_4\times D_5\times F_5$, $D_{10}^2.C_2$, $D_4\times D_5^2$, $D_4\times C_5:F_5$, $D_{20}:C_{20}$, $D_{10}:(C_4\times D_5)$, $D_{10}^2.C_2$, $C_5:D_4\times F_5$

Order 640: $D_5\times D_4^2$

Order 600: $C_{10}\times A_5$ x 2, $D_5\times A_5$ x 2, $C_5:S_5$ x 2, $C_5\times S_5$ x 2

Order 480: $D_{12}:D_{10}$ x 12, $C_2^2\times S_5$ x 7, $D_{10}\times S_4$ x 5, $C_2^2:S_5$ x 4, $C_4\times S_5$ x 4, $D_4\times A_5$ x 3, $C_4:S_5$ x 3, $D_{10}:S_4$, $D_{10}:S_4$, $D_{10}.S_4$, $(C_2\times C_{10}):D_{12}$, $C_5:(C_4\times S_4)$, $C_5:C_4\times S_4$, $C_2\times C_4\times A_5$, $C_2^2.S_5$, $C_{15}:C_2^5$, $C_2\times A_4\times D_{10}$, $C_2^3:D_{30}$, $C_2\times C_{10}\times S_4$, $C_2^3\times A_5$, $D_4\times D_{30}$, $C_{60}:C_2^3$, $C_{60}:C_2^3$, $C_2^4:C_{30}$, $D_{30}:D_4$, $D_{30}:C_2^3$, $D_{30}:C_2^3$, $D_{12}:D_{10}$, $D_6\times D_{20}$, $D_{10}\times D_{12}$, $C_{60}:C_2^3$, $A_4\times C_5:D_4$, $A_4\times D_{20}$, $C_4\times D_5\times A_4$, $C_5:\GL(2,\mathbb{Z}/4)$, $C_{20}:S_4$, $C_{20}:S_4$, $C_5\times \GL(2,\mathbb{Z}/4)$, $C_{20}:S_4$, $C_{20}\times S_4$

Order 400: $D_{10}\times F_5$ x 6, $C_{20}:D_{10}$ x 2, $D_{10}:D_{10}$ x 2, $D_5\times D_{20}$ x 2, $D_{10}:F_5$ x 2, $C_5^2:C_4^2$ x 2, $D_{10}^2$, $C_{10}^2:C_4$, $D_{10}:C_{20}$, $C_{20}:D_{10}$, $D_{10}:D_{10}$, $C_{20}:D_{10}$, $C_4\times D_5^2$, $C_{10}^2:C_4$, $C_{20}:F_5$, $C_{20}:F_5$, $D_{10}:C_{20}$, $C_{20}:C_{20}$, $C_{20}\times F_5$, $D_{10}.D_{10}$

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$

Order 320: $C_{20}:C_2^4$ x 3, $C_2^4:D_{10}$ x 3, $D_{20}:D_4$ x 3, $C_4:D_4\times D_5$ x 3, $C_2^4:D_{10}$ x 2, $C_2^4:D_{10}$ x 2, $D_4\times D_{20}$ x 2, $C_4^2:D_{10}$ x 2, $D_{10}.C_2^4$, $C_5\times D_4^2$, $C_4^2:D_{10}$, $C_4^2:D_{10}$

Order 300: $C_5\times A_5$

Order 240: $D_6\times D_{10}$ x 14, $C_2\times S_5$ x 10, $D_6:D_{10}$ x 8, $D_6:D_{10}$ x 8, $D_6:D_{10}$ x 8, $C_{12}:D_{10}$ x 8, $D_4\times D_{15}$ x 6, $C_{20}:D_6$ x 6, $C_{12}:D_{10}$ x 6, $C_{12}:D_{10}$ x 6, $S_3\times D_{20}$ x 6, $D_5\times D_{12}$ x 6, $D_5\times S_4$ x 4, $C_2^2\times A_5$ x 4, $A_4\times D_{10}$ x 3, $C_{10}:S_4$ x 3, $C_{10}\times S_4$ x 3, $C_4\times A_5$ x 2, $A_5:C_4$ x 2, $C_2^2\times D_{30}$, $C_{15}:C_2^4$, $C_{15}:C_2^4$, $C_2^3:C_{30}$, $D_4\times C_{30}$, $C_{30}:D_4$, $C_2\times D_{60}$, $C_4\times D_{30}$, $C_{30}:D_4$, $C_{10}\times D_{12}$, $D_6\times C_{20}$, $C_{30}:D_4$, $C_6\times D_{20}$, $C_{12}\times D_{10}$, $A_4\times C_{20}$, $C_{10}:D_{12}$, $C_6:D_{20}$, $C_{30}:D_4$, $D_{30}:C_4$, $D_6.D_{10}$, $D_{10}.D_6$, $A_4\times C_5:C_4$, $C_{10}.S_4$, $A_4:C_{20}$

Order 200: $D_5\times D_{10}$ x 6, $D_5\times F_5$ x 5, $C_{10}:F_5$ x 4, $C_{10}\times F_5$ x 4, $C_{10}\times D_{10}$ x 2, $C_{10}\wr C_2$ x 2, $C_5\times D_{20}$ x 2, $D_5\times C_{20}$ x 2, $C_5:D_{20}$ x 2, $C_{10}.D_{10}$ x 2, $C_{10}:D_{10}$, $D_4\times C_5^2$, $C_{10}^2:C_2$, $C_5:D_{20}$, $C_{20}:D_5$, $D_{10}:D_5$, $C_{10}.D_{10}$

Order 192: $D_4\times S_4$ x 6, $C_2^3\times S_4$, $C_{12}:C_2^4$, $C_2^5:C_6$, $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^3:D_{12}$, $C_2^4.D_6$

Order 160: $D_4\times D_{10}$ x 29, $D_4\times F_5$ x 6, $D_{10}:D_4$ x 6, $C_2^2:D_{20}$ x 6, $D_{10}.D_4$ x 5, $C_2^3:D_{10}$ x 4, $C_{20}:D_4$ x 4, $C_2^3.D_{10}$ x 4, $D_{10}:D_4$ x 4, $C_{20}:C_2^3$ x 3, $C_2^2\times D_{20}$ x 3, $C_{20}:C_2^3$ x 3, $C_{20}:D_4$ x 3, $C_{20}:D_4$ x 3, $C_{20}:D_4$ x 3, $C_{20}:D_4$ x 3, $C_4:D_{20}$ x 3, $C_2^3.D_{10}$ x 3, $C_4\times D_{20}$ x 2, $C_2^3\times D_{10}$ x 2, $C_2^4:C_{10}$ x 2, $D_4\times C_{20}$ x 2, $C_2^4:D_5$ x 2, $C_2^3.D_{10}$ x 2, $D_{20}:C_4$ x 2, $D_{10}.D_4$ x 2, $C_4:D_{20}$, $D_5\times C_4^2$, $C_2^3\times F_5$, $C_2^3:F_5$, $D_{10}.D_4$, $C_{10}:C_4^2$, $C_{20}:D_4$

Order 128: $C_2\times D_4^2$

Order 120: $S_3\times D_{10}$ x 34, $C_{10}\times D_6$ x 8, $C_2\times D_{30}$ x 7, $C_6\times D_{10}$ x 7, $C_6.D_{10}$ x 4, $C_6.D_{10}$ x 4, $C_2\times A_5$ x 4, $S_5$ x 4, $C_{15}:D_4$ x 4, $C_4\times D_{15}$ x 4, $C_{15}:D_4$ x 4, $S_3\times C_{20}$ x 4, $C_{15}:D_4$ x 4, $D_5\times C_{12}$ x 4, $C_5:D_{12}$ x 4, $C_3:D_{20}$ x 4, $C_{15}:D_4$ x 4, $D_{15}:C_4$ x 4, $D_4\times C_{15}$ x 3, $D_{60}$ x 3, $C_5\times D_{12}$ x 3, $C_3\times D_{20}$ x 3, $C_{10}\times A_4$ x 2, $D_5\times A_4$ x 2, $C_5:S_4$ x 2, $C_5\times S_4$ x 2, $C_2^2\times C_{30}$, $C_2\times C_{60}$, $C_{30}:C_4$, $C_6:C_{20}$, $C_{10}:C_{12}$

Order 100: $C_5\times D_{10}$ x 7, $D_5^2$ x 6, $C_5\times F_5$ x 3, $C_5:D_{10}$ x 3, $C_5:F_5$ x 3, $C_5:C_{20}$ x 2, $C_5\times C_{20}$, $C_5^2:C_4$, $C_{10}^2$

Order 96: $D_4\times D_6$ x 18, $C_2^2\times S_4$ x 7, $\GL(2,\mathbb{Z}/4)$ x 4, $C_4\times S_4$ x 4, $D_4\times A_4$ x 3, $C_4:S_4$ x 3, $C_2^3\times D_6$, $C_2^3\times A_4$, $C_{12}:C_2^3$, $C_2^3:D_6$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$, $C_2^3:C_{12}$, $C_2^2.S_4$

Order 80: $D_4\times D_5$ x 49, $C_{10}:D_4$ x 25, $C_2^2\times D_{10}$ x 20, $C_4\times D_{10}$ x 18, $D_4\times C_{10}$ x 16, $C_2\times D_{20}$ x 15, $D_{10}:C_4$ x 10, $C_2^2\times F_5$ x 8, $C_2^2:C_{20}$ x 5, $C_{10}.D_4$ x 5, $D_{10}:C_4$ x 4, $C_4\times F_5$ x 4, $C_2^2\times C_{20}$ x 3, $C_2^2.D_{10}$ x 3, $C_{20}:C_4$ x 3, $C_{10}.D_4$ x 3, $C_{20}:C_4$ x 3, $C_2^3\times C_{10}$ x 2, $C_4:C_{20}$ x 2, $C_{20}:C_4$ x 2, $C_4\times C_{20}$

Order 64: $D_4^2$ x 12, $D_4\times C_2^3$ x 3, $C_2^3:D_4$ x 3, $C_2^3:D_4$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$

Order 60: $S_3\times D_5$ x 20, $S_3\times C_{10}$ x 13, $D_{30}$ x 10, $C_3\times D_{10}$ x 10, $C_2\times C_{30}$ x 4, $C_{60}$ x 2, $C_{15}:C_4$ x 2, $C_5:C_{12}$ x 2, $C_3:C_{20}$ x 2, $C_5\times A_4$, $A_5$

Order 50: $C_5\times D_5$ x 5, $C_5\times C_{10}$ x 2, $C_5:D_5$ x 2

Order 48: $S_3\times D_4$ x 40, $C_2^2\times D_6$ x 16, $C_6:D_4$ x 12, $C_4\times D_6$ x 12, $C_2\times S_4$ x 10, $C_6\times D_4$ x 9, $C_2\times D_{12}$ x 9, $C_2^2\times A_4$ x 4, $C_4\times A_4$ x 2, $A_4:C_4$ x 2, $C_2^3\times C_6$, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 40: $C_2\times D_{10}$ x 56, $C_5:D_4$ x 28, $C_4\times D_5$ x 21, $C_5\times D_4$ x 19, $D_{20}$ x 17, $C_2\times F_5$ x 13, $C_2^2\times C_{10}$ x 12, $C_2\times C_{20}$ x 10, $C_{10}:C_4$ x 10

Order 32: $C_2^2\times D_4$ x 38, $C_4:D_4$ x 20, $C_2^2\wr C_2$ x 16, $C_4\times D_4$ x 14, $C_4:D_4$ x 6, $C_2^3:C_4$ x 5, $C_2^3\times C_4$ x 3, $C_2^5$ x 2, $C_2^2.D_4$ x 2, $C_2\times C_4^2$

Order 30: $C_5\times S_3$ x 5, $C_{30}$ x 4, $D_{15}$ x 4, $C_3\times D_5$ x 4

Order 25: $C_5^2$

Order 24: $C_2\times D_6$ x 49, $C_3:D_4$ x 16, $C_4\times S_3$ x 16, $D_{12}$ x 10, $C_3\times D_4$ x 10, $C_2^2\times C_6$ x 8, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6, $C_2\times A_4$ x 4, $S_4$ x 4

Order 20: $D_{10}$ x 49, $C_2\times C_{10}$ x 22, $C_{20}$ x 9, $C_5:C_4$ x 9, $F_5$ x 6

Order 16: $C_2\times D_4$ x 95, $C_2^2\times C_4$ x 23, $C_2^4$ x 21, $C_2^2:C_4$ x 20, $C_4:C_4$ x 7, $C_4^2$ x 4

Order 15: $C_{15}$

Order 12: $D_6$ x 43, $C_2\times C_6$ x 14, $C_{12}$ x 4, $C_3:C_4$ x 4, $A_4$

Order 10: $C_{10}$ x 13, $D_5$ x 13

Order 8: $C_2^3$ x 58, $D_4$ x 51, $C_2\times C_4$ x 34

Order 6: $S_3$ x 9, $C_6$ x 8

Order 5: $C_5$ x 3

Order 4: $C_2^2$ x 50, $C_4$ x 12

Order 3: $C_3$

Order 2: $C_2$ x 13

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 9600: $D_4\times D_5\times S_5$ ($C_1$)

Order 4800: $C_5:D_4.S_5$ ($C_2$) x 2, $C_5:D_4.S_5$ ($C_2$) x 2, $C_2^2\times D_5.S_5$ ($C_2$) x 2, $D_5.C_2^2:S_5$ ($C_2$) x 2, $D_{20}.S_5$ ($C_2$), $D_{20}.S_5$ ($C_2$), $(C_5\times D_4).S_5$ ($C_2$), $(C_4\times D_5).S_5$ ($C_2$), $(D_4\times D_5).A_5$ ($C_2$), $(C_5\times D_4).S_5$ ($C_2$), $(C_4\times D_5).S_5$ ($C_2$)

Order 2400: $C_2\times D_5.S_5$ ($C_2^2$) x 7, $D_{10}.S_5$ ($C_2^2$) x 2, $C_5.C_2^2:S_5$ ($C_2^2$) x 2, $C_5.C_2^2:S_5$ ($C_2^2$) x 2, $C_2^2\times C_5.S_5$ ($C_2^2$) x 2, $C_5:C_4.S_5$ ($C_2^2$) x 2, $(C_2^2\times D_5).A_5$ ($C_2^2$) x 2, $C_5:D_4.A_5$ ($C_2^2$) x 2, $D_{10}.S_5$ ($C_2^2$) x 2, $(C_2\times C_{10}).S_5$ ($C_2^2$) x 2, $C_5:C_4.S_5$ ($C_2^2$), $C_{20}.S_5$ ($C_2^2$), $D_{20}.A_5$ ($C_2^2$), $(C_5\times D_4).A_5$ ($C_2^2$), $C_{20}.S_5$ ($C_2^2$), $C_{20}.S_5$ ($C_2^2$), $C_{20}.S_5$ ($C_2^2$), $C_5:C_4.S_5$ ($C_2^2$), $(C_4\times D_5).A_5$ ($C_2^2$), $D_{10}.S_5$ ($C_2^2$)

Order 1200: $D_5\times S_5$ ($D_4$) x 4, $D_{10}\times A_5$ ($C_2^3$) x 3, $C_{10}\times S_5$ ($C_2^3$) x 3, $C_{10}:S_5$ ($C_2^3$) x 3, $C_2\times C_{10}\times A_5$ ($C_2^3$) x 2, $C_{20}\times A_5$ ($C_2^3$), $C_5:C_4\times A_5$ ($C_2^3$), $C_{10}.S_5$ ($C_2^3$), $A_5:C_{20}$ ($C_2^3$)

Order 960: $D_4\times S_5$ ($D_5$)

Order 600: $D_5\times A_5$ ($C_2\times D_4$) x 2, $C_5:S_5$ ($C_2\times D_4$) x 2, $C_5\times S_5$ ($C_2\times D_4$) x 2, $C_{10}\times A_5$ ($C_2^4$)

Order 480: $C_2^2:S_5$ ($D_{10}$) x 2, $C_2^2\times S_5$ ($D_{10}$) x 2, $D_4\times A_5$ ($D_{10}$), $C_4:S_5$ ($D_{10}$), $C_4\times S_5$ ($D_{10}$)

Order 300: $C_5\times A_5$ ($C_2^2\times D_4$)

Order 240: $C_2\times S_5$ ($C_2\times D_{10}$) x 3, $C_2^2\times A_5$ ($C_2\times D_{10}$) x 2, $C_4\times A_5$ ($C_2\times D_{10}$), $A_5:C_4$ ($C_2\times D_{10}$)

Order 120: $S_5$ ($D_4\times D_5$) x 2, $C_2\times A_5$ ($C_2^2\times D_{10}$)

Order 80: $D_4\times D_5$ ($S_5$)

Order 60: $A_5$ ($D_4\times D_{10}$)

Order 40: $C_5:D_4$ ($C_2\times S_5$) x 2, $C_2\times D_{10}$ ($C_2\times S_5$) x 2, $D_{20}$ ($C_2\times S_5$), $C_4\times D_5$ ($C_2\times S_5$), $C_5\times D_4$ ($C_2\times S_5$)

Order 20: $D_{10}$ ($C_2^2\times S_5$) x 3, $C_2\times C_{10}$ ($C_2^2\times S_5$) x 2, $C_{20}$ ($C_2^2\times S_5$), $C_5:C_4$ ($C_2^2\times S_5$)

Order 10: $D_5$ ($D_4\times S_5$) x 2, $C_{10}$ ($C_2^3\times S_5$)

Order 8: $D_4$ ($D_5\times S_5$)

Order 5: $C_5$ ($C_2\times D_4\times S_5$)

Order 4: $C_2^2$ ($D_{10}.S_5$) x 2, $C_4$ ($D_{10}.S_5$)

Order 2: $C_2$ ($D_{10}.(C_2\times S_5)$)

Order 1: $C_1$ ($D_4\times D_5\times S_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 9600: $D_4\times D_5\times S_5$ ($C_1$)

Order 4800: $D_{20}.S_5$ ($C_2$), $D_{20}.S_5$ ($C_2$), $C_5:D_4.S_5$ ($C_2$), $C_5:D_4.S_5$ ($C_2$), $(C_5\times D_4).S_5$ ($C_2$), $C_2^2\times D_5.S_5$ ($C_2$), $D_5.C_2^2:S_5$ ($C_2$), $(C_4\times D_5).S_5$ ($C_2$), $(D_4\times D_5).A_5$ ($C_2$), $(C_5\times D_4).S_5$ ($C_2$), $(C_4\times D_5).S_5$ ($C_2$)

Order 2400: $C_2\times D_5.S_5$ ($C_2^2$) x 4, $D_{10}.S_5$ ($C_2^2$), $C_5.C_2^2:S_5$ ($C_2^2$), $C_5:C_4.S_5$ ($C_2^2$), $C_{20}.S_5$ ($C_2^2$), $C_5.C_2^2:S_5$ ($C_2^2$), $D_{20}.A_5$ ($C_2^2$), $(C_5\times D_4).A_5$ ($C_2^2$), $C_2^2\times C_5.S_5$ ($C_2^2$), $C_{20}.S_5$ ($C_2^2$), $C_5:C_4.S_5$ ($C_2^2$), $(C_2^2\times D_5).A_5$ ($C_2^2$), $C_{20}.S_5$ ($C_2^2$), $C_5:D_4.A_5$ ($C_2^2$), $C_{20}.S_5$ ($C_2^2$), $C_5:C_4.S_5$ ($C_2^2$), $(C_4\times D_5).A_5$ ($C_2^2$), $D_{10}.S_5$ ($C_2^2$), $D_{10}.S_5$ ($C_2^2$), $(C_2\times C_{10}).S_5$ ($C_2^2$)

Order 1200: $D_{10}\times A_5$ ($C_2^3$) x 2, $C_{10}\times S_5$ ($C_2^3$) x 2, $C_{10}:S_5$ ($C_2^3$) x 2, $C_2\times C_{10}\times A_5$ ($C_2^3$), $D_5\times S_5$ ($D_4$), $C_{20}\times A_5$ ($C_2^3$), $C_5:C_4\times A_5$ ($C_2^3$), $C_{10}.S_5$ ($C_2^3$), $A_5:C_{20}$ ($C_2^3$)

Order 960: $D_4\times S_5$ ($D_5$)

Order 600: $C_{10}\times A_5$ ($C_2^4$), $D_5\times A_5$ ($C_2\times D_4$), $C_5:S_5$ ($C_2\times D_4$), $C_5\times S_5$ ($C_2\times D_4$)

Order 480: $D_4\times A_5$ ($D_{10}$), $C_2^2:S_5$ ($D_{10}$), $C_4:S_5$ ($D_{10}$), $C_4\times S_5$ ($D_{10}$), $C_2^2\times S_5$ ($D_{10}$)

Order 300: $C_5\times A_5$ ($C_2^2\times D_4$)

Order 240: $C_2\times S_5$ ($C_2\times D_{10}$) x 2, $C_4\times A_5$ ($C_2\times D_{10}$), $A_5:C_4$ ($C_2\times D_{10}$), $C_2^2\times A_5$ ($C_2\times D_{10}$)

Order 120: $C_2\times A_5$ ($C_2^2\times D_{10}$), $S_5$ ($D_4\times D_5$)

Order 80: $D_4\times D_5$ ($S_5$)

Order 60: $A_5$ ($D_4\times D_{10}$)

Order 40: $C_5:D_4$ ($C_2\times S_5$), $D_{20}$ ($C_2\times S_5$), $C_4\times D_5$ ($C_2\times S_5$), $C_2\times D_{10}$ ($C_2\times S_5$), $C_5\times D_4$ ($C_2\times S_5$)

Order 20: $D_{10}$ ($C_2^2\times S_5$) x 2, $C_2\times C_{10}$ ($C_2^2\times S_5$), $C_{20}$ ($C_2^2\times S_5$), $C_5:C_4$ ($C_2^2\times S_5$)

Order 10: $C_{10}$ ($C_2^3\times S_5$), $D_5$ ($D_4\times S_5$)

Order 8: $D_4$ ($D_5\times S_5$)

Order 5: $C_5$ ($C_2\times D_4\times S_5$)

Order 4: $C_2^2$ ($D_{10}.S_5$), $C_4$ ($D_{10}.S_5$)

Order 2: $C_2$ ($D_{10}.(C_2\times S_5)$)

Order 1: $C_1$ ($D_4\times D_5\times S_5$)

Series

Derived series

$D_4\times D_5\times S_5$

$\rhd$

$D_4\times D_5\times S_5$

$\rhd$

$C_{10}\times A_5$

$\rhd$

$C_{10}\times A_5$

$\rhd$

$A_5$

$\rhd$

$A_5$

Chief series

$D_4\times D_5\times S_5$

$\rhd$

$D_4\times D_5\times S_5$

$\rhd$

$(D_4\times D_5).A_5$

$\rhd$

$(D_4\times D_5).A_5$

$\rhd$

$D_4\times D_5$

$\rhd$

$D_4\times D_5$

$\rhd$

$C_5\times D_4$

$\rhd$

$C_5\times D_4$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_2\times C_{10}$

$\rhd$

$C_{10}$

$\rhd$

$C_{10}$

$\rhd$

$C_5$

$\rhd$

$C_5$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$D_4\times D_5\times S_5$

$\rhd$

$D_4\times D_5\times S_5$

$\rhd$

$C_{10}\times A_5$

$\rhd$

$C_{10}\times A_5$

$\rhd$

$C_5\times A_5$

$\rhd$

$C_5\times A_5$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

$\lhd$

$D_4$

$\lhd$

$D_4$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $140 \times 140$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $105 \times 105$ rational character table.