Abstract group 86400.u: $S_4\times A_5^2$

• Label: 86400.u
• Order: \( 2^{7} \cdot 3^{3} \cdot 5^{2} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{3} \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $14$
• Trans deg.: $100$
• Rank: $2$

Group information

Description:	$S_4\times A_5^2$
Order:	\(86400\)\(\medspace = 2^{7} \cdot 3^{3} \cdot 5^{2} \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$S_4\times S_5\wr C_2$, of order \(691200\)\(\medspace = 2^{10} \cdot 3^{3} \cdot 5^{2} \)
Composition factors:	$C_2$ x 3, $C_3$, $A_5$ x 2
Derived length:	$3$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30	60
Elements	1	2559	3968	1536	624	16800	12816	6240	13632	8064	14400	5760	86400
Conjugacy classes	1	11	7	4	8	17	28	5	16	12	12	4	125
Divisions	1	11	7	4	4	17	14	5	8	6	6	2	85
Autjugacy classes	1	8	5	3	2	10	7	3	4	3	3	1	50

Dimension	1	2	3	4	5	6	8	9	10	12	15	16	18	20	24	25	27	30	32	36	40	45	48	50	54	60	72	75	90
Irr. complex chars.	2	1	10	4	4	4	2	16	2	12	12	2	4	4	4	2	8	4	1	8	2	8	2	1	0	4	0	2	0	125
Irr. rational chars.	2	1	2	4	4	4	2	0	2	6	4	2	8	4	4	2	0	4	1	2	2	0	4	1	4	6	4	2	4	85

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$100$
Rank:	$2$
Inequivalent generating pairs:	$1539$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	27	27	48
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $14$ $\langle(2,4)(3,5), (2,3)(4,5), (2,5,4), (1,12,14,11,13)(3,4)(6,7,8,9,10), (4,5), (1,12)(2,5,3,4)(6,9)(7,10)(11,13)\rangle$
Direct product:	$S_4$ $\, \times\, $ $A_5$ ${}^2$
Semidirect product:	$(A_4\times A_5^2)$ $\,\rtimes\,$ $C_2$	$A_4$ $\,\rtimes\,$ $(C_2\times A_5^2)$	$(C_2^2\times A_5^2)$ $\,\rtimes\,$ $S_3$	$C_2^2$ $\,\rtimes\,$ $(S_3\times A_5^2)$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{2} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 1256232 subgroups in 3278 conjugacy classes, 16 normal (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $S_4\times A_5^2$
Commutator:	$G' \simeq$ $A_4\times A_5^2$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_4\times A_5^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2$	$G/\operatorname{Fit} \simeq$ $S_3\times A_5^2$
Radical:	$R \simeq$ $S_4$	$G/R \simeq$ $A_5^2$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times A_5^2$	$G/\operatorname{soc} \simeq$ $S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times C_2^4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$
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 86400: $S_4\times A_5^2$

Order 43200: $A_4\times A_5^2$

Order 28800: $D_4\times A_5^2$

Order 21600: $S_3\times A_5^2$

Order 17280: $A_4\times S_4\times A_5$ x 2

Order 14400: $D_5\times S_4\times A_5$ x 2, $C_2^2\times A_5^2$ x 2, $C_4\times A_5^2$

Order 10800: $A_5\times \GL(2,4)$

Order 8640: $A_5\times A_4^2$ x 2, $S_3\times S_4\times A_5$ x 2

Order 7200: $D_5\times A_4\times A_5$ x 2, $C_5\times S_4\times A_5$ x 2, $A_5\times C_5:S_4$ x 2, $C_2\times A_5^2$ x 2

Order 5760: $D_4\times A_4\times A_5$ x 2, $C_2^2\times S_4\times A_5$ x 2

Order 4800: $D_4\times D_5\times A_5$ x 2

Order 4320: $S_3\times A_4\times A_5$ x 4, $\GL(2,4):S_4$ x 2, $S_4\times \GL(2,4)$ x 2

Order 3600: $C_5\times A_4\times A_5$ x 2, $\GL(2,4):D_{10}$ x 2, $A_5^2$

Order 3456: $S_4\times A_4^2$

Order 2880: $C_2^4:\GL(2,4)$ x 6, $C_2\times S_4\times A_5$ x 4, $C_2^4.\GL(2,4)$ x 2, $A_4^2:D_{10}$ x 2, $C_4\times A_4\times A_5$ x 2, $S_3\times D_4\times A_5$ x 2

Order 2400: $C_2\times D_{10}\times A_5$ x 4, $C_5:D_4\times A_5$ x 4, $C_4\times D_5\times A_5$ x 2, $D_{20}\times A_5$ x 2, $C_5\times D_4\times A_5$ x 2, $S_4\times D_5^2$

Order 2160: $A_4\times \GL(2,4)$ x 4, $S_3^2\times A_5$ x 2

Order 1920: $C_2^2\times D_4\times A_5$ x 2

Order 1800: $D_{15}\times A_5$ x 2, $D_5\times \GL(2,4)$ x 2, $\GL(2,4):C_{10}$ x 2

Order 1728: $A_4^2:D_6$ x 2, $A_4^3$

Order 1440: $S_4\times A_5$ x 8, $\GL(2,4):C_2^3$ x 6, $C_2^3:\GL(2,4)$ x 6, $\GL(2,4):D_4$ x 4, $D_5\times A_4^2$ x 2, $A_4^2:D_5$ x 2, $A_4^2:C_{10}$ x 2, $S_3\times D_5\times S_4$ x 2, $D_4\times \GL(2,4)$ x 2, $D_{12}\times A_5$ x 2, $C_4\times S_3\times A_5$ x 2

Order 1200: $D_{10}\times A_5$ x 10, $C_2\times C_{10}\times A_5$ x 4, $A_4:D_5^2$ x 2, $C_{10}^2:D_6$ x 2, $C_{20}\times A_5$ x 2, $C_5:C_4\times A_5$ x 2, $A_4\times D_5^2$, $A_4:D_5^2$, $C_{10}^2:D_6$

Order 1152: $C_2^2:A_4\times S_4$ x 2, $A_4^2:C_2^3$ x 2, $D_4\times A_4^2$

Order 1080: $S_3\times \GL(2,4)$ x 4, $\GL(2,4):S_3$ x 2

Order 960: $C_2\times D_4\times A_5$ x 8, $C_2^4\times A_5$ x 4, $C_2\times D_{10}\times S_4$ x 2, $D_4\times D_5\times A_4$ x 2, $C_4\times C_2^2\times A_5$ x 2

Order 900: $C_{15}\times A_5$ x 2

Order 864: $S_3\times A_4^2$ x 3, $A_4^2:S_3$ x 2, $A_4^2:C_6$ x 2, $S_4\times S_3^2$

Order 800: $D_4\times D_5^2$

Order 720: $D_6\times A_5$ x 14, $A_4\times A_5$ x 12, $C_2^2\times \GL(2,4)$ x 6, $S_3\times D_5\times A_4$ x 4, $C_5\times A_4^2$ x 2, $D_5\times C_3:S_4$ x 2, $D_{15}:S_4$ x 2, $S_4\times D_{15}$ x 2, $S_3\times C_5:S_4$ x 2, $C_5\times S_3\times S_4$ x 2, $C_3\times D_5\times S_4$ x 2, $C_{12}\times A_5$ x 2, $\GL(2,4):C_4$ x 2

Order 600: $D_5\times A_5$ x 6, $C_{10}\times A_5$ x 4, $C_{10}^2:C_6$ x 2, $C_5^2:S_4$ x 2, $C_{10}^2:C_6$, $C_5^2:S_4$, $S_4\times C_5^2$, $S_3\times D_5^2$

Order 576: $C_2^2:A_4^2$ x 4, $C_2^2\times A_4^2$ x 4, $A_4^2:C_2^2$ x 4, $C_2^2:A_4^2$ x 2, $C_2^2:D_6^2$ x 2, $S_3\times D_4\times A_4$ x 2, $C_4\times A_4^2$

Order 540: $C_3\times \GL(2,4)$ x 2

Order 480: $D_4\times A_5$ x 14, $C_2^3\times A_5$ x 14, $D_{10}\times S_4$ x 8, $C_2\times A_4\times D_{10}$ x 6, $C_2\times C_4\times A_5$ x 4, $A_4\times C_5:D_4$ x 4, $D_5\times C_2^2:A_4$ x 2, $C_2^3:D_{30}$ x 2, $C_2\times C_{10}\times S_4$ x 2, $C_2^4:C_{30}$ x 2, $D_{12}:D_{10}$ x 2, $A_4\times D_{20}$ x 2, $C_4\times D_5\times A_4$ x 2

Order 432: $C_3\times A_4^2$ x 3, $A_4\times S_3^2$ x 3, $A_4:S_3^2$ x 2, $C_6^2:D_6$ x 2, $A_4:S_3^2$, $C_6^2:D_6$

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

Order 384: $C_2^6:C_6$ x 2, $C_2^6:C_6$ x 2, $C_2^4\times S_4$

Order 360: $S_3\times A_5$ x 14, $C_6\times A_5$ x 6, $A_4\times D_{15}$ x 4, $C_5\times S_3\times A_4$ x 4, $C_3\times D_5\times A_4$ x 4, $C_{15}:S_4$ x 2, $C_{15}:S_4$ x 2, $C_{15}:S_4$ x 2, $C_{15}\times S_4$ x 2, $D_5\times S_3^2$ x 2

Order 320: $C_{20}:C_2^4$ x 2

Order 300: $D_5\times D_{15}$ x 2, $C_{15}:D_{10}$ x 2, $C_5\times A_5$ x 2, $A_4\times C_5^2$, $C_{15}:D_{10}$, $C_{15}:D_{10}$, $C_3\times D_5^2$

Order 288: $A_4\times S_4$ x 10, $C_2\times A_4\times D_6$ x 8, $D_6\times S_4$ x 8, $A_4\times C_3:D_4$ x 4, $S_3\times C_2^2:A_4$ x 4, $C_2\times A_4^2$ x 4, $C_2^3.C_6^2$ x 2, $A_4\times D_{12}$ x 2, $C_4\times S_3\times A_4$ x 2, $C_2\times C_6:S_4$ x 2, $C_2\times C_6\times S_4$ x 2, $D_4\times S_3^2$

Order 240: $C_2^2\times A_5$ x 28, $D_5\times S_4$ x 16, $A_4\times D_{10}$ x 14, $C_4\times A_5$ x 6, $C_2^3:C_{30}$ x 6, $D_6\times D_{10}$ x 6, $C_{10}:S_4$ x 4, $C_{10}\times S_4$ x 4, $D_6:D_{10}$ x 4, $D_6:D_{10}$ x 4, $D_6:D_{10}$ x 4, $C_2^4:C_{15}$ x 2, $D_4\times D_{15}$ x 2, $C_{20}:D_6$ x 2, $C_{12}:D_{10}$ x 2, $A_4\times C_{20}$ x 2, $C_{12}:D_{10}$ x 2, $S_3\times D_{20}$ x 2, $D_5\times D_{12}$ x 2, $C_{12}:D_{10}$ x 2, $A_4\times C_5:C_4$ x 2

Order 216: $C_6^2:C_6$ x 6, $C_6^2:C_6$ x 3, $C_3^2:S_4$ x 2, $C_3^2:S_4$, $C_3^2\times S_4$, $S_3^3$

Order 200: $D_5\times D_{10}$ x 11, $C_{10}\times D_{10}$ 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: $C_2^5:C_6$ x 8, $C_2^2\wr C_3$ x 6, $C_2^3\times S_4$ x 6, $A_4\times C_2^4$ x 5, $C_2^4:A_4$ x 2, $C_{12}:C_2^4$ x 2, $C_2^4:C_{12}$ x 2, $C_2^4:C_{12}$ x 2

Order 180: $\GL(2,4)$ x 8, $A_4\times C_{15}$ x 4, $S_3\times D_{15}$ x 4, $C_{15}:D_6$ x 4, $C_{15}:D_6$ x 2, $C_5\times S_3^2$ x 2, $C_{15}:D_6$ x 2

Order 160: $D_4\times D_{10}$ x 16, $C_2^3\times D_{10}$ x 4, $C_2^3:D_{10}$ x 4, $C_{20}:C_2^3$ x 2, $C_2^2\times D_{20}$ x 2, $C_{20}:C_2^3$ x 2

Order 150: $D_5\times C_{15}$ x 2, $C_5\times D_{15}$ x 2, $C_{15}:D_5$, $C_5:D_{15}$, $S_3\times C_5^2$

Order 144: $A_4\times D_6$ x 18, $A_4^2$ x 17, $S_3\times S_4$ x 15, $C_2^2:C_6^2$ x 8, $C_2^4:C_3^2$ x 4, $D_6^2$ x 4, $C_6:S_4$ x 4, $C_6\times S_4$ x 4, $D_6:D_6$ x 4, $C_{12}:D_6$ x 2, $C_{12}\times A_4$ x 2, $D_6:D_6$ x 2, $S_3\times D_{12}$ x 2, $A_4\times C_3:C_4$ x 2, $C_{12}:D_6$, $C_{12}:D_6$, $C_4\times S_3^2$

Order 128: $D_4\times C_2^4$

Order 120: $S_3\times D_{10}$ x 30, $D_5\times A_4$ x 18, $C_2\times A_5$ x 14, $C_5:S_4$ x 8, $C_5\times S_4$ x 8, $C_2\times D_{30}$ x 6, $C_{10}\times D_6$ x 6, $C_6\times D_{10}$ x 6, $C_{10}\times A_4$ x 6, $C_{15}:D_4$ x 4, $C_{15}:D_4$ x 4, $C_{15}:D_4$ x 4, $C_5:D_{12}$ x 4, $C_3:D_{20}$ x 4, $C_{15}:D_4$ x 4, $C_6.D_{10}$ x 2, $C_6.D_{10}$ x 2, $D_4\times C_{15}$ x 2, $D_{60}$ x 2, $C_4\times D_{15}$ x 2, $C_5\times D_{12}$ x 2, $S_3\times C_{20}$ x 2, $C_3\times D_{20}$ x 2, $D_5\times C_{12}$ x 2, $D_{15}:C_4$ x 2

Order 108: $C_3^2\times A_4$ x 3, $C_3:S_3^2$ x 3, $C_3\times S_3^2$ x 3, $C_3:S_3^2$

Order 100: $D_5^2$ x 11, $C_5\times D_{10}$ x 10, $C_5:D_{10}$ 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: $C_2^2\times S_4$ x 22, $C_2^3\times A_4$ x 17, $D_4\times D_6$ x 16, $D_4\times A_4$ x 16, $C_2^3:A_4$ x 6, $C_2^3\times D_6$ x 5, $C_2^3:D_6$ x 4, $C_2^3:C_{12}$ x 4, $C_{12}:C_2^3$ x 2, $C_2^2\times D_{12}$ x 2, $C_{12}:C_2^3$ x 2

Order 90: $C_3\times D_{15}$ x 4, $S_3\times C_{15}$ x 4, $C_3:D_{15}$ x 2, $C_{15}:S_3$ x 2, $C_3^2\times D_5$ x 2

Order 80: $D_4\times D_5$ x 46, $C_2^2\times D_{10}$ x 30, $C_{10}:D_4$ x 16, $D_4\times C_{10}$ x 8, $C_2\times D_{20}$ x 8, $C_4\times D_{10}$ x 8, $C_2^3\times C_{10}$ x 4, $C_2^2\times C_{20}$ x 2, $C_2^2.D_{10}$ x 2

Order 75: $C_5\times C_{15}$

Order 72: $S_3\times A_4$ x 27, $S_3\times D_6$ x 19, $C_6\times D_6$ x 8, $C_6\times A_4$ x 8, $C_3:S_4$ x 8, $C_3\times S_4$ x 7, $C_6:D_6$ x 4, $C_6\wr C_2$ x 4, $C_3:D_{12}$ x 4, $C_6^2:C_2$ x 2, $C_3\times D_{12}$ x 2, $S_3\times C_{12}$ x 2, $D_6:S_3$ x 2, $C_6.D_6$ x 2, $D_4\times C_3^2$, $C_3:D_{12}$, $C_{12}:S_3$, $C_6.D_6$

Order 64: $D_4\times C_2^3$ x 12, $C_2^6$ x 2, $C_2^4\times C_4$

Order 60: $S_3\times D_5$ x 38, $D_{30}$ x 14, $S_3\times C_{10}$ x 14, $C_3\times D_{10}$ x 14, $C_5\times A_4$ x 12, $C_2\times C_{30}$ x 6, $A_5$ x 4, $C_{60}$ x 2, $C_{15}:C_4$ x 2, $C_5:C_{12}$ x 2, $C_3:C_{20}$ x 2

Order 54: $C_3^2:C_6$ x 3, $S_3\times C_3^2$ x 3, $C_3^2:S_3$

Order 50: $C_5\times D_5$ x 6, $C_5:D_5$ x 3, $C_5\times C_{10}$ x 2

Order 48: $C_2^2\times A_4$ x 45, $S_3\times D_4$ x 45, $C_2^2\times D_6$ x 36, $C_2\times S_4$ x 20, $C_2^2:A_4$ x 16, $C_6:D_4$ x 16, $C_6\times D_4$ x 8, $C_2\times D_{12}$ x 8, $C_4\times D_6$ x 8, $C_4\times A_4$ x 8, $C_2^3\times C_6$ x 5, $C_2^2\times C_{12}$ x 2, $C_6.C_2^3$ x 2

Order 45: $C_3\times C_{15}$ x 2

Order 40: $C_2\times D_{10}$ x 90, $C_5:D_4$ x 28, $D_{20}$ x 14, $C_4\times D_5$ x 14, $C_2^2\times C_{10}$ x 14, $C_5\times D_4$ x 14, $C_2\times C_{20}$ x 4, $C_{10}:C_4$ x 4

Order 36: $S_3^2$ x 26, $C_3\times A_4$ x 22, $C_6\times S_3$ x 18, $C_6:S_3$ x 9, $C_6^2$ x 4, $C_3:C_{12}$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$

Order 32: $C_2^2\times D_4$ x 72, $C_2^5$ x 17, $C_2^3\times C_4$ x 6

Order 30: $D_{15}$ x 14, $C_3\times D_5$ x 14, $C_5\times S_3$ x 14, $C_{30}$ x 6

Order 27: $C_3^3$

Order 25: $C_5^2$

Order 24: $C_2\times D_6$ x 110, $C_2\times A_4$ x 27, $C_3:D_4$ x 26, $C_2^2\times C_6$ x 17, $D_{12}$ x 13, $C_4\times S_3$ x 13, $C_3\times D_4$ x 13, $S_4$ x 9, $C_2\times C_{12}$ x 4, $C_6:C_4$ x 4

Order 20: $D_{10}$ x 74, $C_2\times C_{10}$ x 28, $C_{20}$ x 6, $C_5:C_4$ x 6

Order 18: $C_3\times S_3$ x 20, $C_3:S_3$ x 11, $C_3\times C_6$ x 4

Order 16: $C_2\times D_4$ x 112, $C_2^4$ x 86, $C_2^2\times C_4$ x 20

Order 15: $C_{15}$ x 8

Order 12: $D_6$ x 89, $C_2\times C_6$ x 35, $A_4$ x 22, $C_{12}$ x 5, $C_3:C_4$ x 5

Order 10: $D_5$ x 18, $C_{10}$ x 14

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 136, $D_4$ x 36, $C_2\times C_4$ x 16

Order 6: $S_3$ x 24, $C_6$ x 17

Order 5: $C_5$ x 4

Order 4: $C_2^2$ x 70, $C_4$ x 4

Order 3: $C_3$ x 7

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 86400: $S_4\times A_5^2$

Order 43200: $A_4\times A_5^2$

Order 28800: $D_4\times A_5^2$

Order 21600: $S_3\times A_5^2$

Order 17280: $A_4\times S_4\times A_5$

Order 14400: $C_2^2\times A_5^2$ x 2, $D_5\times S_4\times A_5$, $C_4\times A_5^2$

Order 10800: $A_5\times \GL(2,4)$

Order 8640: $A_5\times A_4^2$, $S_3\times S_4\times A_5$

Order 7200: $C_2\times A_5^2$ x 2, $D_5\times A_4\times A_5$, $C_5\times S_4\times A_5$, $A_5\times C_5:S_4$

Order 5760: $D_4\times A_4\times A_5$, $C_2^2\times S_4\times A_5$

Order 4800: $D_4\times D_5\times A_5$

Order 4320: $S_3\times A_4\times A_5$ x 2, $\GL(2,4):S_4$, $S_4\times \GL(2,4)$

Order 3600: $C_5\times A_4\times A_5$, $\GL(2,4):D_{10}$, $A_5^2$

Order 3456: $S_4\times A_4^2$

Order 2880: $C_2^4:\GL(2,4)$ x 3, $C_2\times S_4\times A_5$ x 2, $C_2^4.\GL(2,4)$, $A_4^2:D_{10}$, $C_4\times A_4\times A_5$, $S_3\times D_4\times A_5$

Order 2400: $C_2\times D_{10}\times A_5$ x 2, $C_5:D_4\times A_5$ x 2, $C_4\times D_5\times A_5$, $D_{20}\times A_5$, $C_5\times D_4\times A_5$, $S_4\times D_5^2$

Order 2160: $A_4\times \GL(2,4)$ x 2, $S_3^2\times A_5$

Order 1920: $C_2^2\times D_4\times A_5$

Order 1800: $D_{15}\times A_5$, $D_5\times \GL(2,4)$, $\GL(2,4):C_{10}$

Order 1728: $A_4^3$, $A_4^2:D_6$

Order 1440: $S_4\times A_5$ x 4, $\GL(2,4):C_2^3$ x 3, $C_2^3:\GL(2,4)$ x 3, $\GL(2,4):D_4$ x 2, $D_5\times A_4^2$, $A_4^2:D_5$, $A_4^2:C_{10}$, $S_3\times D_5\times S_4$, $D_4\times \GL(2,4)$, $D_{12}\times A_5$, $C_4\times S_3\times A_5$

Order 1200: $D_{10}\times A_5$ x 5, $C_2\times C_{10}\times A_5$ x 2, $A_4\times D_5^2$, $A_4:D_5^2$, $A_4:D_5^2$, $C_{10}^2:D_6$, $C_{10}^2:D_6$, $C_{20}\times A_5$, $C_5:C_4\times A_5$

Order 1152: $C_2^2:A_4\times S_4$, $A_4^2:C_2^3$, $D_4\times A_4^2$

Order 1080: $S_3\times \GL(2,4)$ x 2, $\GL(2,4):S_3$

Order 960: $C_2\times D_4\times A_5$ x 4, $C_2^4\times A_5$ x 2, $C_2\times D_{10}\times S_4$, $D_4\times D_5\times A_4$, $C_4\times C_2^2\times A_5$

Order 900: $C_{15}\times A_5$

Order 864: $S_3\times A_4^2$ x 2, $A_4^2:S_3$, $A_4^2:C_6$, $S_4\times S_3^2$

Order 800: $D_4\times D_5^2$

Order 720: $D_6\times A_5$ x 7, $A_4\times A_5$ x 6, $C_2^2\times \GL(2,4)$ x 3, $S_3\times D_5\times A_4$ x 2, $C_5\times A_4^2$, $D_5\times C_3:S_4$, $D_{15}:S_4$, $S_4\times D_{15}$, $S_3\times C_5:S_4$, $C_5\times S_3\times S_4$, $C_3\times D_5\times S_4$, $C_{12}\times A_5$, $\GL(2,4):C_4$

Order 600: $D_5\times A_5$ x 3, $C_{10}\times A_5$ x 2, $C_{10}^2:C_6$, $C_{10}^2:C_6$, $C_5^2:S_4$, $C_5^2:S_4$, $S_4\times C_5^2$, $S_3\times D_5^2$

Order 576: $C_2^2\times A_4^2$ x 3, $C_2^2:A_4^2$ x 2, $A_4^2:C_2^2$ x 2, $C_2^2:A_4^2$, $C_2^2:D_6^2$, $S_3\times D_4\times A_4$, $C_4\times A_4^2$

Order 540: $C_3\times \GL(2,4)$

Order 480: $C_2^3\times A_5$ x 7, $D_4\times A_5$ x 6, $D_{10}\times S_4$ x 4, $C_2\times A_4\times D_{10}$ x 3, $C_2\times C_4\times A_5$ x 2, $A_4\times C_5:D_4$ x 2, $D_5\times C_2^2:A_4$, $C_2^3:D_{30}$, $C_2\times C_{10}\times S_4$, $C_2^4:C_{30}$, $D_{12}:D_{10}$, $A_4\times D_{20}$, $C_4\times D_5\times A_4$

Order 432: $C_3\times A_4^2$ x 2, $A_4\times S_3^2$ x 2, $A_4:S_3^2$, $A_4:S_3^2$, $C_6^2:D_6$, $C_6^2:D_6$

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

Order 384: $C_2^4\times S_4$, $C_2^6:C_6$, $C_2^6:C_6$

Order 360: $S_3\times A_5$ x 7, $C_6\times A_5$ x 3, $A_4\times D_{15}$ x 2, $C_5\times S_3\times A_4$ x 2, $C_3\times D_5\times A_4$ x 2, $C_{15}:S_4$, $C_{15}:S_4$, $C_{15}:S_4$, $C_{15}\times S_4$, $D_5\times S_3^2$

Order 320: $C_{20}:C_2^4$

Order 300: $A_4\times C_5^2$, $C_{15}:D_{10}$, $D_5\times D_{15}$, $C_{15}:D_{10}$, $C_{15}:D_{10}$, $C_3\times D_5^2$, $C_5\times A_5$

Order 288: $A_4\times S_4$ x 5, $C_2\times A_4\times D_6$ x 4, $D_6\times S_4$ x 4, $C_2\times A_4^2$ x 3, $A_4\times C_3:D_4$ x 2, $S_3\times C_2^2:A_4$ x 2, $C_2^3.C_6^2$, $D_4\times S_3^2$, $A_4\times D_{12}$, $C_4\times S_3\times A_4$, $C_2\times C_6:S_4$, $C_2\times C_6\times S_4$

Order 240: $C_2^2\times A_5$ x 14, $A_4\times D_{10}$ x 7, $D_5\times S_4$ x 7, $C_4\times A_5$ x 3, $C_2^3:C_{30}$ x 3, $D_6\times D_{10}$ x 3, $C_{10}:S_4$ x 2, $C_{10}\times S_4$ x 2, $D_6:D_{10}$ x 2, $D_6:D_{10}$ x 2, $D_6:D_{10}$ x 2, $C_2^4:C_{15}$, $D_4\times D_{15}$, $C_{20}:D_6$, $C_{12}:D_{10}$, $A_4\times C_{20}$, $C_{12}:D_{10}$, $S_3\times D_{20}$, $D_5\times D_{12}$, $C_{12}:D_{10}$, $A_4\times C_5:C_4$

Order 216: $C_6^2:C_6$ x 3, $C_6^2:C_6$ x 2, $C_3^2:S_4$, $C_3^2:S_4$, $C_3^2\times S_4$, $S_3^3$

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

Order 192: $C_2^3\times S_4$ x 4, $C_2^5:C_6$ x 4, $C_2^2\wr C_3$ x 3, $A_4\times C_2^4$ x 3, $C_2^4:A_4$, $C_{12}:C_2^4$, $C_2^4:C_{12}$, $C_2^4:C_{12}$

Order 180: $\GL(2,4)$ x 4, $A_4\times C_{15}$ x 2, $S_3\times D_{15}$ x 2, $C_{15}:D_6$ x 2, $C_{15}:D_6$, $C_5\times S_3^2$, $C_{15}:D_6$

Order 160: $D_4\times D_{10}$ x 8, $C_2^3\times D_{10}$ x 2, $C_2^3:D_{10}$ x 2, $C_{20}:C_2^3$, $C_2^2\times D_{20}$, $C_{20}:C_2^3$

Order 150: $C_{15}:D_5$, $D_5\times C_{15}$, $C_5:D_{15}$, $C_5\times D_{15}$, $S_3\times C_5^2$

Order 144: $A_4\times D_6$ x 9, $A_4^2$ x 9, $S_3\times S_4$ x 7, $C_2^2:C_6^2$ x 4, $D_6^2$ x 3, $C_2^4:C_3^2$ x 2, $C_6:S_4$ x 2, $C_6\times S_4$ x 2, $D_6:D_6$ x 2, $D_6:D_6$ x 2, $C_{12}:D_6$, $C_{12}:D_6$, $C_{12}\times A_4$, $C_{12}:D_6$, $S_3\times D_{12}$, $C_4\times S_3^2$, $A_4\times C_3:C_4$

Order 128: $D_4\times C_2^4$

Order 120: $S_3\times D_{10}$ x 15, $D_5\times A_4$ x 9, $C_2\times A_5$ x 7, $C_5:S_4$ x 4, $C_5\times S_4$ x 4, $C_2\times D_{30}$ x 3, $C_{10}\times D_6$ x 3, $C_6\times D_{10}$ x 3, $C_{10}\times A_4$ x 3, $C_{15}:D_4$ x 2, $C_{15}:D_4$ x 2, $C_{15}:D_4$ x 2, $C_5:D_{12}$ x 2, $C_3:D_{20}$ x 2, $C_{15}:D_4$ x 2, $C_6.D_{10}$, $C_6.D_{10}$, $D_4\times C_{15}$, $D_{60}$, $C_4\times D_{15}$, $C_5\times D_{12}$, $S_3\times C_{20}$, $C_3\times D_{20}$, $D_5\times C_{12}$, $D_{15}:C_4$

Order 108: $C_3^2\times A_4$ x 2, $C_3:S_3^2$ x 2, $C_3\times S_3^2$ x 2, $C_3:S_3^2$

Order 100: $D_5^2$ x 8, $C_5:D_{10}$ x 5, $C_5\times D_{10}$ x 5, $C_{10}^2$ x 2, $C_5\times C_{20}$, $C_5^2:C_4$, $C_5:C_{20}$

Order 96: $C_2^2\times S_4$ x 11, $C_2^3\times A_4$ x 9, $D_4\times D_6$ x 8, $D_4\times A_4$ x 7, $C_2^3\times D_6$ x 3, $C_2^3:A_4$ x 3, $C_2^3:D_6$ x 2, $C_2^3:C_{12}$ x 2, $C_{12}:C_2^3$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$

Order 90: $C_3\times D_{15}$ x 2, $S_3\times C_{15}$ x 2, $C_3:D_{15}$, $C_{15}:S_3$, $C_3^2\times D_5$

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

Order 75: $C_5\times C_{15}$

Order 72: $S_3\times A_4$ x 14, $S_3\times D_6$ x 12, $C_3:S_4$ x 5, $C_6\times D_6$ x 4, $C_6\times A_4$ x 4, $C_3\times S_4$ x 4, $C_6:D_6$ x 3, $C_6^2:C_2$ x 2, $C_6\wr C_2$ x 2, $C_3:D_{12}$ x 2, $D_4\times C_3^2$, $C_3:D_{12}$, $C_{12}:S_3$, $C_3\times D_{12}$, $S_3\times C_{12}$, $D_6:S_3$, $C_6.D_6$, $C_6.D_6$

Order 64: $D_4\times C_2^3$ x 8, $C_2^6$ x 2, $C_2^4\times C_4$

Order 60: $S_3\times D_5$ x 18, $D_{30}$ x 7, $S_3\times C_{10}$ x 7, $C_3\times D_{10}$ x 7, $C_5\times A_4$ x 6, $C_2\times C_{30}$ x 3, $A_5$ x 2, $C_{60}$, $C_{15}:C_4$, $C_5:C_{12}$, $C_3:C_{20}$

Order 54: $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2, $C_3^2:S_3$

Order 50: $C_5:D_5$ x 3, $C_5\times D_5$ x 3, $C_5\times C_{10}$ x 2

Order 48: $C_2^2\times A_4$ x 23, $C_2^2\times D_6$ x 19, $S_3\times D_4$ x 16, $C_2\times S_4$ x 10, $C_6:D_4$ x 8, $C_2^2:A_4$ x 7, $C_6\times D_4$ x 4, $C_2\times D_{12}$ x 4, $C_4\times D_6$ x 4, $C_4\times A_4$ x 4, $C_2^3\times C_6$ x 3, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 45: $C_3\times C_{15}$

Order 40: $C_2\times D_{10}$ x 41, $C_5:D_4$ x 12, $C_2^2\times C_{10}$ x 7, $D_{20}$ x 6, $C_4\times D_5$ x 6, $C_5\times D_4$ x 6, $C_2\times C_{20}$ x 2, $C_{10}:C_4$ x 2

Order 36: $S_3^2$ x 15, $C_3\times A_4$ x 12, $C_6\times S_3$ x 9, $C_6:S_3$ x 7, $C_6^2$ x 3, $C_3\times C_{12}$, $C_3^2:C_4$, $C_3:C_{12}$

Order 32: $C_2^2\times D_4$ x 30, $C_2^5$ x 12, $C_2^3\times C_4$ x 4

Order 30: $D_{15}$ x 7, $C_3\times D_5$ x 7, $C_5\times S_3$ x 7, $C_{30}$ x 3

Order 27: $C_3^3$

Order 25: $C_5^2$

Order 24: $C_2\times D_6$ x 52, $C_2\times A_4$ x 14, $C_3:D_4$ x 12, $C_2^2\times C_6$ x 9, $D_{12}$ x 6, $C_4\times S_3$ x 6, $S_4$ x 6, $C_3\times D_4$ x 6, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2

Order 20: $D_{10}$ x 34, $C_2\times C_{10}$ x 14, $C_{20}$ x 3, $C_5:C_4$ x 3

Order 18: $C_3\times S_3$ x 11, $C_3:S_3$ x 8, $C_3\times C_6$ x 3

Order 16: $C_2^4$ x 45, $C_2\times D_4$ x 39, $C_2^2\times C_4$ x 10

Order 15: $C_{15}$ x 4

Order 12: $D_6$ x 44, $C_2\times C_6$ x 19, $A_4$ x 12, $C_{12}$ x 3, $C_3:C_4$ x 3

Order 10: $D_5$ x 9, $C_{10}$ x 7

Order 9: $C_3^2$ x 5

Order 8: $C_2^3$ x 62, $D_4$ x 15, $C_2\times C_4$ x 8

Order 6: $S_3$ x 15, $C_6$ x 10

Order 5: $C_5$ x 2

Order 4: $C_2^2$ x 38, $C_4$ x 3

Order 3: $C_3$ x 5

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 86400: $S_4\times A_5^2$ ($C_1$)

Order 43200: $A_4\times A_5^2$ ($C_2$)

Order 14400: $C_2^2\times A_5^2$ ($S_3$)

Order 3600: $A_5^2$ ($S_4$)

Order 1440: $S_4\times A_5$ ($A_5$) x 2

Order 720: $A_4\times A_5$ ($C_2\times A_5$) x 2

Order 240: $C_2^2\times A_5$ ($S_3\times A_5$) x 2

Order 60: $A_5$ ($S_4\times A_5$) x 2

Order 24: $S_4$ ($A_5^2$)

Order 12: $A_4$ ($C_2\times A_5^2$)

Order 4: $C_2^2$ ($S_3\times A_5^2$)

Order 1: $C_1$ ($S_4\times A_5^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 86400: $S_4\times A_5^2$ ($C_1$)

Order 43200: $A_4\times A_5^2$ ($C_2$)

Order 14400: $C_2^2\times A_5^2$ ($S_3$)

Order 3600: $A_5^2$ ($S_4$)

Order 1440: $S_4\times A_5$ ($A_5$)

Order 720: $A_4\times A_5$ ($C_2\times A_5$)

Order 240: $C_2^2\times A_5$ ($S_3\times A_5$)

Order 60: $A_5$ ($S_4\times A_5$)

Order 24: $S_4$ ($A_5^2$)

Order 12: $A_4$ ($C_2\times A_5^2$)

Order 4: $C_2^2$ ($S_3\times A_5^2$)

Order 1: $C_1$ ($S_4\times A_5^2$)

Series

Derived series

$S_4\times A_5^2$

$\rhd$

$S_4\times A_5^2$

$\rhd$

$A_4\times A_5^2$

$\rhd$

$A_4\times A_5^2$

$\rhd$

$C_2^2\times A_5^2$

$\rhd$

$C_2^2\times A_5^2$

$\rhd$

$A_5^2$

$\rhd$

$A_5^2$

Chief series

$S_4\times A_5^2$

$\rhd$

$S_4\times A_5^2$

$\rhd$

$S_4\times A_5$

$\rhd$

$S_4\times A_5$

$\rhd$

$S_4$

$\rhd$

$S_4$

$\rhd$

$A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$S_4\times A_5^2$

$\rhd$

$S_4\times A_5^2$

$\rhd$

$A_4\times A_5^2$

$\rhd$

$A_4\times A_5^2$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 4 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 $125 \times 125$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $85 \times 85$ rational character table.