Abstract group 9600.bz: $C_2^2\times F_5\times S_5$

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

Group information

Description:	$C_2^2\times F_5\times S_5$
Order:	\(9600\)\(\medspace = 2^{7} \cdot 3 \cdot 5^{2} \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$F_5.S_5\times C_2^2:S_4$, of order \(230400\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5^{2} \)
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
Elements	1	623	20	2960	124	940	1252	1600	80	1440	560	9600
Conjugacy classes	1	23	1	40	3	15	21	16	1	12	7	140
Divisions	1	23	1	24	3	15	21	8	1	8	7	112
Autjugacy classes	1	9	1	10	3	5	8	4	1	3	2	47

Dimension	1	2	4	5	6	8	10	12	16	20	24
Irr. complex chars.	32	0	40	32	16	0	0	0	8	8	4	140
Irr. rational chars.	16	8	24	16	8	8	8	4	8	8	4	112

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$100$
Rank:	$4$
Inequivalent generating quadruples:	$11098143840$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $14$ $\langle(4,5)(6,9,8,10,7), (1,11)(4,5)(6,8,7,9,10)(12,13,14), (6,9,7,10), (2,3)(4,5), (4,5)(6,8,9,7)(11,14), (2,3)(6,7)(9,10), (4,5)\rangle$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $F_5$ $\, \times\, $ $S_5$
Semidirect product:	$(D_{10}\times S_5)$ $\,\rtimes\,$ $C_4$	$D_{10}$ $\,\rtimes\,$ $(C_4\times S_5)$	$(F_5\times A_5)$ $\,\rtimes\,$ $C_2^3$	$(A_5:F_5)$ $\,\rtimes\,$ $C_2^3$	all 27
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_5\times S_5)$ . $C_2^3$	$D_5$ . $(C_2^3\times S_5)$	$(D_5\times A_5)$ . $C_2^4$	$(D_{10}\times S_5)$ . $C_2^2$	all 9
Aut. group:	$\Aut(C_5:C_4\times \SL(2,5))$	$\Aut(\SL(2,5).D_{10})$	$\Aut(C_5:D_4\times A_5)$	$\Aut(D_{10}\times S_5)$	all 21

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

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$1$

Subgroups

There are 152892 subgroups in 4469 conjugacy classes, 166 normal (20 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $F_5\times S_5$
Commutator:	$G' \simeq$ $C_5\times A_5$	$G/G' \simeq$ $C_2^3\times C_4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^2\times F_5\times S_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{10}$	$G/\operatorname{Fit} \simeq$ $C_4\times S_5$
Radical:	$R \simeq$ $C_2^2\times F_5$	$G/R \simeq$ $S_5$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{10}\times A_5$	$G/\operatorname{soc} \simeq$ $C_2\times C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_2^3$
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

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 9600: $C_2^2\times F_5\times S_5$

Order 4800: $C_2\times F_5\times S_5$ x 12, $C_2\times D_{10}\times S_5$, $C_2^2\times F_5\times A_5$, $C_2\times D_{10}.S_5$

Order 2400: $F_5\times S_5$ x 16, $D_{10}\times S_5$ x 12, $C_2\times F_5\times A_5$ x 6, $D_{10}.S_5$ x 6, $C_2\times C_{10}:S_5$, $C_2\times C_{10}\times S_5$, $C_2\times D_{10}\times A_5$

Order 1920: $C_4\times C_2^2\times S_5$, $C_2^2\times F_5\times S_4$

Order 1600: $C_2^2\times F_5^2$

Order 1200: $D_5\times S_5$ x 16, $D_{10}\times A_5$ x 6, $C_{10}\times S_5$ x 6, $C_{10}:S_5$ x 6, $F_5\times A_5$ x 4, $A_5:F_5$ x 4, $C_2\times C_{10}\times A_5$

Order 960: $C_2\times F_5\times S_4$ x 12, $C_2\times C_4\times S_5$ x 12, $C_2^2\times D_6\times F_5$, $C_2\times D_{10}\times S_4$, $C_2^2\times A_4\times F_5$, $C_2^2\times A_4:F_5$, $C_2^3\times S_5$, $C_4\times C_2^2\times A_5$, $C_2^3.S_5$

Order 800: $C_2\times F_5^2$ x 12, $D_{10}^2.C_2$ x 2, $D_{10}^2.C_2$

Order 640: $D_{10}.C_2^5$

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: $C_2\times D_6\times F_5$ x 28, $C_4\times S_5$ x 16, $F_5\times S_4$ x 16, $C_2^2\times S_5$ x 14, $D_{10}\times S_4$ x 12, $C_2\times C_4\times A_5$ x 6, $C_2^2.S_5$ x 6, $C_2\times A_4\times F_5$ x 6, $D_{10}.S_4$ x 6, $C_{15}:C_2^5$, $C_2^2\times C_{30}:C_4$, $C_2^2\times C_6\times F_5$, $C_2\times A_4\times D_{10}$, $C_2^3:D_{30}$, $C_2\times C_{10}\times S_4$, $C_2^3\times A_5$

Order 400: $D_{10}\times F_5$ x 24, $F_5^2$ x 16, $D_{10}:F_5$ x 12, $C_{10}^2:C_4$ x 2, $D_{10}:C_{20}$ x 2, $D_{10}^2$, $C_{10}^2:C_4$, $C_{10}^2:C_4$

Order 384: $C_2^5.D_6$

Order 320: $D_{10}.C_2^4$ x 24, $C_2^4\times F_5$ x 2, $C_2^4:F_5$ x 2, $D_{10}:C_4^2$ x 2, $C_{20}:C_2^4$, $D_{10}.C_2^4$

Order 300: $C_5\times A_5$

Order 240: $D_6\times F_5$ x 112, $D_6\times D_{10}$ x 28, $C_2\times S_5$ x 28, $D_5\times S_4$ x 16, $D_{10}.D_6$ x 14, $D_{10}:C_{12}$ x 14, $C_2^2\times A_5$ x 7, $A_4\times D_{10}$ x 6, $C_{10}:S_4$ x 6, $C_{10}\times S_4$ x 6, $C_4\times A_5$ x 4, $A_5:C_4$ x 4, $A_4\times F_5$ x 4, $A_4:F_5$ x 4, $C_2^2\times D_{30}$, $C_{15}:C_2^4$, $C_{15}:C_2^4$, $C_2^3:C_{30}$

Order 200: $D_5\times F_5$ x 32, $D_5:F_5$ x 16, $D_5\times D_{10}$ x 12, $C_{10}:F_5$ x 12, $C_{10}\times F_5$ x 12, $C_{10}:F_5$ x 6, $C_{10}:F_5$ x 6, $C_{10}\times D_{10}$ x 2, $C_{10}:D_{10}$

Order 192: $C_2^4.D_6$ x 12, $C_2^3\times S_4$, $C_{12}:C_2^4$, $C_2^4:C_{12}$, $C_2^3.S_4$

Order 160: $D_4\times F_5$ x 64, $C_2^3\times F_5$ x 38, $D_4\times D_{10}$ x 24, $C_2^3:F_5$ x 24, $C_{10}:C_4^2$ x 24, $D_{10}.D_4$ x 12, $C_2^3\times D_{10}$ x 2, $C_2^3:D_{10}$ x 2, $C_{20}:C_2^3$ x 2, $C_{20}:C_2^3$, $C_2^2\times D_{20}$

Order 128: $C_4^2:C_2^3$

Order 120: $S_3\times D_{10}$ x 112, $S_3\times F_5$ x 64, $C_{30}:C_4$ x 28, $C_6\times F_5$ x 28, $C_2\times D_{30}$ x 14, $C_{10}\times D_6$ x 14, $C_6\times D_{10}$ x 14, $S_5$ x 8, $C_2\times A_5$ x 7, $D_5\times A_4$ x 4, $C_5:S_4$ x 4, $C_5\times S_4$ x 4, $C_{10}\times A_4$ x 3, $C_2^2\times C_{30}$

Order 100: $D_5^2$ x 16, $C_5\times D_{10}$ x 12, $C_5\times F_5$ x 8, $C_5:F_5$ x 8, $C_5:D_{10}$ x 6, $C_5:F_5$ x 4, $C_5:F_5$ x 4, $C_{10}^2$

Order 96: $C_{12}:C_2^3$ x 28, $C_4\times S_4$ x 16, $C_2^2\times S_4$ x 14, $C_2^3:C_{12}$ x 6, $C_2^2.S_4$ x 6, $C_2^3\times D_6$, $C_2^3\times A_4$, $C_2^3\times C_{12}$, $C_6.C_2^4$

Order 80: $C_2^2\times F_5$ x 156, $D_4\times D_5$ x 64, $C_2^2\times D_{10}$ x 36, $D_{10}:C_4$ x 32, $C_4\times F_5$ x 32, $C_{10}:D_4$ x 24, $C_4\times D_{10}$ x 24, $C_{20}:C_4$ x 16, $D_4\times C_{10}$ x 12, $C_2\times D_{20}$ x 12, $C_2^3\times C_{10}$ x 2, $C_2^2\times C_{20}$ x 2, $C_2^2.D_{10}$ x 2

Order 64: $C_4^2:C_2^2$ x 24, $C_2^4\times C_4$ x 2, $C_2^4:C_4$ x 2, $D_4\times C_2^3$, $C_2^3.D_4$, $C_2^2\times C_4^2$

Order 60: $S_3\times D_5$ x 64, $D_{30}$ x 28, $S_3\times C_{10}$ x 28, $C_3\times D_{10}$ x 28, $C_{15}:C_4$ x 8, $C_3\times F_5$ x 8, $C_2\times C_{30}$ x 7, $C_5\times A_4$, $A_5$

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

Order 48: $C_4\times D_6$ x 112, $C_2^2\times D_6$ x 30, $C_2\times S_4$ x 28, $C_2^2\times C_{12}$ x 14, $C_6.C_2^3$ x 14, $C_2^2\times A_4$ x 7, $C_4\times A_4$ x 4, $A_4:C_4$ x 4, $C_2^3\times C_6$

Order 40: $C_2\times F_5$ x 164, $C_2\times D_{10}$ x 132, $C_5:D_4$ x 32, $C_4\times D_5$ x 32, $C_2^2\times C_{10}$ x 18, $D_{20}$ x 16, $C_5\times D_4$ x 16, $C_2\times C_{20}$ x 12, $C_{10}:C_4$ x 12

Order 32: $C_4\times D_4$ x 64, $C_2^3\times C_4$ x 37, $C_2^2\times D_4$ x 28, $C_2^3:C_4$ x 24, $C_2^2.D_4$ x 12, $C_2\times C_4^2$ x 12, $C_2^5$ x 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 140, $C_4\times S_3$ x 64, $C_2\times C_{12}$ x 28, $C_6:C_4$ x 28, $C_2^2\times C_6$ x 15, $S_4$ x 8, $C_2\times A_4$ x 7

Order 20: $D_{10}$ x 120, $C_2\times C_{10}$ x 36, $F_5$ x 36, $C_{20}$ x 8, $C_5:C_4$ x 8

Order 16: $C_2^2\times C_4$ x 141, $C_2\times D_4$ x 112, $C_2^4$ x 37, $C_2^2:C_4$ x 32, $C_4:C_4$ x 16, $C_4^2$ x 16

Order 15: $C_{15}$

Order 12: $D_6$ x 120, $C_2\times C_6$ x 35, $C_{12}$ x 8, $C_3:C_4$ x 8, $A_4$

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

Order 8: $C_2^3$ x 134, $C_2\times C_4$ x 130, $D_4$ x 64

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

Order 5: $C_5$ x 3

Order 4: $C_2^2$ x 114, $C_4$ x 24

Order 3: $C_3$

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 9600: $C_2^2\times F_5\times S_5$

Order 4800: $C_2\times D_{10}\times S_5$, $C_2^2\times F_5\times A_5$, $C_2\times D_{10}.S_5$, $C_2\times F_5\times S_5$

Order 2400: $D_{10}\times S_5$ x 2, $C_2\times C_{10}:S_5$, $C_2\times C_{10}\times S_5$, $C_2\times D_{10}\times A_5$, $C_2\times F_5\times A_5$, $D_{10}.S_5$, $F_5\times S_5$

Order 1920: $C_4\times C_2^2\times S_5$, $C_2^2\times F_5\times S_4$

Order 1600: $C_2^2\times F_5^2$

Order 1200: $D_{10}\times A_5$ x 2, $D_5\times S_5$ x 2, $C_2\times C_{10}\times A_5$, $C_{10}\times S_5$, $C_{10}:S_5$, $F_5\times A_5$, $A_5:F_5$

Order 960: $C_2^2\times D_6\times F_5$, $C_2\times D_{10}\times S_4$, $C_2^2\times A_4\times F_5$, $C_2^2\times A_4:F_5$, $C_2\times F_5\times S_4$, $C_2^3\times S_5$, $C_4\times C_2^2\times A_5$, $C_2^3.S_5$, $C_2\times C_4\times S_5$

Order 800: $D_{10}^2.C_2$ x 2, $D_{10}^2.C_2$, $C_2\times F_5^2$

Order 640: $D_{10}.C_2^5$

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

Order 480: $C_2\times D_6\times F_5$ x 6, $C_2^2\times S_5$ x 4, $D_{10}\times S_4$ x 2, $C_2\times C_4\times A_5$, $C_2^2.S_5$, $C_4\times S_5$, $C_{15}:C_2^5$, $C_2^2\times C_{30}:C_4$, $C_2^2\times C_6\times F_5$, $C_2\times A_4\times D_{10}$, $C_2^3:D_{30}$, $C_2\times C_{10}\times S_4$, $C_2\times A_4\times F_5$, $D_{10}.S_4$, $F_5\times S_4$, $C_2^3\times A_5$

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

Order 384: $C_2^5.D_6$

Order 320: $C_2^4\times F_5$ x 2, $C_2^4:F_5$ x 2, $D_{10}.C_2^4$ x 2, $D_{10}:C_4^2$ x 2, $C_{20}:C_2^4$, $D_{10}.C_2^4$

Order 300: $C_5\times A_5$

Order 240: $D_6\times D_{10}$ x 8, $D_6\times F_5$ x 8, $C_2\times S_5$ x 4, $D_{10}.D_6$ x 3, $D_{10}:C_{12}$ x 3, $C_2^2\times A_5$ x 3, $A_4\times D_{10}$ x 2, $D_5\times S_4$ x 2, $C_4\times A_5$, $A_5:C_4$, $C_2^2\times D_{30}$, $C_{15}:C_2^4$, $C_{15}:C_2^4$, $C_2^3:C_{30}$, $C_{10}:S_4$, $C_{10}\times S_4$, $A_4\times F_5$, $A_4:F_5$

Order 200: $D_5\times D_{10}$ x 4, $D_5\times F_5$ x 4, $C_{10}\times D_{10}$ x 2, $C_{10}:F_5$ x 2, $C_{10}\times F_5$ x 2, $D_5:F_5$ x 2, $C_{10}:D_{10}$, $C_{10}:F_5$, $C_{10}:F_5$

Order 192: $C_2^3\times S_4$, $C_{12}:C_2^4$, $C_2^4:C_{12}$, $C_2^3.S_4$, $C_2^4.D_6$

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

Order 128: $C_4^2:C_2^3$

Order 120: $S_3\times D_{10}$ x 15, $S_3\times F_5$ x 6, $C_2\times D_{30}$ x 4, $C_{10}\times D_6$ x 4, $C_6\times D_{10}$ x 4, $C_{30}:C_4$ x 3, $C_6\times F_5$ x 3, $C_2\times A_5$ x 3, $D_5\times A_4$ x 2, $S_5$ x 2, $C_2^2\times C_{30}$, $C_{10}\times A_4$, $C_5:S_4$, $C_5\times S_4$

Order 100: $D_5^2$ x 5, $C_5\times D_{10}$ x 4, $C_5\times F_5$ x 2, $C_5:D_{10}$ x 2, $C_5:F_5$ x 2, $C_{10}^2$, $C_5:F_5$, $C_5:F_5$

Order 96: $C_{12}:C_2^3$ x 6, $C_2^2\times S_4$ x 4, $C_2^3\times D_6$, $C_2^3\times A_4$, $C_2^3\times C_{12}$, $C_6.C_2^4$, $C_2^3:C_{12}$, $C_2^2.S_4$, $C_4\times S_4$

Order 80: $C_2^2\times F_5$ x 28, $C_2^2\times D_{10}$ x 14, $D_4\times D_5$ x 5, $C_{10}:D_4$ x 4, $C_4\times D_{10}$ x 4, $D_{10}:C_4$ x 3, $C_2^3\times C_{10}$ x 2, $D_4\times C_{10}$ x 2, $C_2^2\times C_{20}$ x 2, $C_2^2.D_{10}$ x 2, $C_2\times D_{20}$ x 2, $C_4\times F_5$ x 2, $C_{20}:C_4$

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

Order 60: $S_3\times D_5$ x 11, $D_{30}$ x 5, $S_3\times C_{10}$ x 5, $C_3\times D_{10}$ x 5, $C_{15}:C_4$ x 2, $C_3\times F_5$ x 2, $C_2\times C_{30}$ x 2, $C_5\times A_4$, $A_5$

Order 50: $C_5\times D_5$ x 4, $C_5:D_5$ x 2, $C_5\times C_{10}$

Order 48: $C_2^2\times D_6$ x 10, $C_4\times D_6$ x 8, $C_2\times S_4$ x 4, $C_2^2\times A_4$ x 3, $C_2^2\times C_{12}$ x 3, $C_6.C_2^3$ x 3, $C_2^3\times C_6$, $C_4\times A_4$, $A_4:C_4$

Order 40: $C_2\times D_{10}$ x 35, $C_2\times F_5$ x 23, $C_2^2\times C_{10}$ x 7, $C_5:D_4$ x 4, $C_4\times D_5$ x 4, $C_2\times C_{20}$ x 2, $C_{10}:C_4$ x 2, $D_{20}$ x 2, $C_5\times D_4$ x 2

Order 32: $C_2^3\times C_4$ x 11, $C_2^2\times D_4$ x 8, $C_2^3:C_4$ x 3, $C_2^5$ x 2, $C_4\times D_4$ x 2, $C_2^2.D_4$, $C_2\times C_4^2$

Order 30: $D_{15}$ x 3, $C_3\times D_5$ x 3, $C_5\times S_3$ x 3, $C_{30}$ x 2

Order 25: $C_5^2$

Order 24: $C_2\times D_6$ x 23, $C_4\times S_3$ x 6, $C_2^2\times C_6$ x 5, $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3, $C_2\times A_4$ x 3, $S_4$ x 2

Order 20: $D_{10}$ x 31, $C_2\times C_{10}$ x 13, $F_5$ x 9, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 16: $C_2^2\times C_4$ x 23, $C_2^4$ x 15, $C_2\times D_4$ x 13, $C_2^2:C_4$ x 3, $C_4:C_4$, $C_4^2$

Order 15: $C_{15}$

Order 12: $D_6$ x 21, $C_2\times C_6$ x 7, $C_{12}$ x 2, $C_3:C_4$ x 2, $A_4$

Order 10: $D_5$ x 11, $C_{10}$ x 8

Order 8: $C_2^3$ x 34, $C_2\times C_4$ x 18, $D_4$ x 8

Order 6: $S_3$ x 6, $C_6$ x 5

Order 5: $C_5$ x 3

Order 4: $C_2^2$ x 29, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 9600: $C_2^2\times F_5\times S_5$ ($C_1$)

Order 4800: $C_2\times F_5\times S_5$ ($C_2$) x 12, $C_2\times D_{10}\times S_5$ ($C_2$), $C_2^2\times F_5\times A_5$ ($C_2$), $C_2\times D_{10}.S_5$ ($C_2$)

Order 2400: $F_5\times S_5$ ($C_2^2$) x 16, $D_{10}\times S_5$ ($C_2^2$) x 6, $D_{10}\times S_5$ ($C_4$) x 6, $C_2\times F_5\times A_5$ ($C_2^2$) x 6, $D_{10}.S_5$ ($C_2^2$) x 6, $C_2\times C_{10}:S_5$ ($C_4$), $C_2\times C_{10}\times S_5$ ($C_4$), $C_2\times D_{10}\times A_5$ ($C_2^2$)

Order 1200: $D_5\times S_5$ ($C_2\times C_4$) x 12, $C_{10}\times S_5$ ($C_2\times C_4$) x 6, $C_{10}:S_5$ ($C_2\times C_4$) x 6, $D_5\times S_5$ ($C_2^3$) x 4, $F_5\times A_5$ ($C_2^3$) x 4, $A_5:F_5$ ($C_2^3$) x 4, $D_{10}\times A_5$ ($C_2^3$) x 3, $D_{10}\times A_5$ ($C_2\times C_4$) x 3, $C_2\times C_{10}\times A_5$ ($C_2\times C_4$)

Order 600: $C_5:S_5$ ($C_2^2\times C_4$) x 4, $C_5\times S_5$ ($C_2^2\times C_4$) x 4, $C_{10}\times A_5$ ($C_2^2\times C_4$) x 3, $D_5\times A_5$ ($C_2^2\times C_4$) x 3, $D_5\times A_5$ ($C_2^4$)

Order 480: $C_2^2\times S_5$ ($F_5$)

Order 300: $C_5\times A_5$ ($C_2^3\times C_4$)

Order 240: $C_2\times S_5$ ($C_2\times F_5$) x 6, $C_2^2\times A_5$ ($C_2\times F_5$)

Order 120: $S_5$ ($C_2^2\times F_5$) x 4, $C_2\times A_5$ ($C_2^2\times F_5$) x 3

Order 80: $C_2^2\times F_5$ ($S_5$)

Order 60: $A_5$ ($C_2^3\times F_5$)

Order 40: $C_2\times F_5$ ($C_2\times S_5$) x 6, $C_2\times D_{10}$ ($C_2\times S_5$)

Order 20: $F_5$ ($C_2^2\times S_5$) x 4, $D_{10}$ ($C_4\times S_5$) x 3, $D_{10}$ ($C_2^2\times S_5$) x 3, $C_2\times C_{10}$ ($C_4\times S_5$)

Order 10: $C_{10}$ ($C_2\times C_4\times S_5$) x 3, $D_5$ ($C_2\times C_4\times S_5$) x 3, $D_5$ ($C_2^3\times S_5$)

Order 5: $C_5$ ($C_4\times C_2^2\times S_5$)

Order 4: $C_2^2$ ($F_5\times S_5$)

Order 2: $C_2$ ($C_2\times F_5\times S_5$) x 3

Order 1: $C_1$ ($C_2^2\times F_5\times S_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 9600: $C_2^2\times F_5\times S_5$ ($C_1$)

Order 4800: $C_2\times D_{10}\times S_5$ ($C_2$), $C_2^2\times F_5\times A_5$ ($C_2$), $C_2\times D_{10}.S_5$ ($C_2$), $C_2\times F_5\times S_5$ ($C_2$)

Order 2400: $C_2\times C_{10}:S_5$ ($C_4$), $C_2\times C_{10}\times S_5$ ($C_4$), $C_2\times D_{10}\times A_5$ ($C_2^2$), $D_{10}\times S_5$ ($C_2^2$), $D_{10}\times S_5$ ($C_4$), $C_2\times F_5\times A_5$ ($C_2^2$), $D_{10}.S_5$ ($C_2^2$), $F_5\times S_5$ ($C_2^2$)

Order 1200: $C_2\times C_{10}\times A_5$ ($C_2\times C_4$), $D_{10}\times A_5$ ($C_2^3$), $D_{10}\times A_5$ ($C_2\times C_4$), $C_{10}\times S_5$ ($C_2\times C_4$), $C_{10}:S_5$ ($C_2\times C_4$), $D_5\times S_5$ ($C_2^3$), $D_5\times S_5$ ($C_2\times C_4$), $F_5\times A_5$ ($C_2^3$), $A_5:F_5$ ($C_2^3$)

Order 600: $C_{10}\times A_5$ ($C_2^2\times C_4$), $D_5\times A_5$ ($C_2^4$), $D_5\times A_5$ ($C_2^2\times C_4$), $C_5:S_5$ ($C_2^2\times C_4$), $C_5\times S_5$ ($C_2^2\times C_4$)

Order 480: $C_2^2\times S_5$ ($F_5$)

Order 300: $C_5\times A_5$ ($C_2^3\times C_4$)

Order 240: $C_2^2\times A_5$ ($C_2\times F_5$), $C_2\times S_5$ ($C_2\times F_5$)

Order 120: $C_2\times A_5$ ($C_2^2\times F_5$), $S_5$ ($C_2^2\times F_5$)

Order 80: $C_2^2\times F_5$ ($S_5$)

Order 60: $A_5$ ($C_2^3\times F_5$)

Order 40: $C_2\times D_{10}$ ($C_2\times S_5$), $C_2\times F_5$ ($C_2\times S_5$)

Order 20: $C_2\times C_{10}$ ($C_4\times S_5$), $D_{10}$ ($C_4\times S_5$), $D_{10}$ ($C_2^2\times S_5$), $F_5$ ($C_2^2\times S_5$)

Order 10: $C_{10}$ ($C_2\times C_4\times S_5$), $D_5$ ($C_2^3\times S_5$), $D_5$ ($C_2\times C_4\times S_5$)

Order 5: $C_5$ ($C_4\times C_2^2\times S_5$)

Order 4: $C_2^2$ ($F_5\times S_5$)

Order 2: $C_2$ ($C_2\times F_5\times S_5$)

Order 1: $C_1$ ($C_2^2\times F_5\times S_5$)

Series

Derived series

$C_2^2\times F_5\times S_5$

$\rhd$

$C_2^2\times F_5\times S_5$

$\rhd$

$C_5\times A_5$

$\rhd$

$C_5\times A_5$

$\rhd$

$A_5$

$\rhd$

$A_5$

Chief series

$C_2^2\times F_5\times S_5$

$\rhd$

$C_2^2\times F_5\times S_5$

$\rhd$

$C_2^2\times F_5\times A_5$

$\rhd$

$C_2^2\times F_5\times A_5$

$\rhd$

$C_2^2\times F_5$

$\rhd$

$C_2^2\times F_5$

$\rhd$

$C_2\times F_5$

$\rhd$

$C_2\times F_5$

$\rhd$

$F_5$

$\rhd$

$F_5$

$\rhd$

$D_5$

$\rhd$

$D_5$

$\rhd$

$C_5$

$\rhd$

$C_5$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_2^2\times F_5\times S_5$

$\rhd$

$C_2^2\times F_5\times S_5$

$\rhd$

$C_5\times A_5$

$\rhd$

$C_5\times A_5$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^2$

Supergroups

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

This group is a maximal quotient of 1 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 $112 \times 112$ rational character table.