Abstract group 57600.bx: $A_5^2:C_2^4$

• Label: 57600.bx
• Order: \( 2^{8} \cdot 3^{2} \cdot 5^{2} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{3} \cdot 5^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $40$
• Rank: $4$

Group information

Description:	$A_5^2:C_2^4$
Order:	\(57600\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5^{2} \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^2:S_4.A_5^2.D_4$, of order \(2764800\)\(\medspace = 2^{12} \cdot 3^{3} \cdot 5^{2} \)
Composition factors:	$C_2$ x 4, $A_5$ x 2
Derived length:	$1$

This group is nonabelian, nonsolvable, and rational.

Group statistics

Order	1	2	3	4	5	6	10	12	15	30
Elements	1	1903	440	13200	624	16520	16272	4800	960	2880	57600
Conjugacy classes	1	23	2	16	3	26	21	4	1	3	100
Divisions	1	23	2	16	3	26	21	4	1	3	100
Autjugacy classes	1	7	2	3	2	7	5	1	1	1	30

Dimension	1	8	10	12	16	18	25	40	48	60
Irr. complex chars.	16	8	8	4	16	16	16	8	4	4	100
Irr. rational chars.	16	8	8	4	16	16	16	8	4	4	100

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$40$
Rank:	$4$
Inequivalent generating quadruples:	$1224602978025$

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 $14$ $\langle(1,3,8,6)(2,5,7)(4,9)(11,12)(13,14), (1,4,8,9)(2,3,5,6)(7,10)(11,12)(13,14) \!\cdots\! \rangle$
Transitive group:	40T18862				more information
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $\POPlus(4,5)$
Semidirect product:	$A_5^2$ $\,\rtimes\,$ $C_2^4$	$(A_5^2:C_2^3)$ $\,\rtimes\,$ $C_2$	$(A_5^2:C_2^3)$ $\,\rtimes\,$ $C_2$	$(C_2\times A_5^2)$ $\,\rtimes\,$ $C_2^3$	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut((C_2\times \SD_{16}).S_5)$	$\Aut((C_2^2\times S_5):D_4)$	$\Aut(C_3^6.C_{36})$	$\Aut(C_3^5.(S_3\times C_{18}))$	all 12

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 1699307 subgroups in 5950 conjugacy classes, 72 normal (6 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $\POPlus(4,5)$
Commutator:	$G' \simeq$ $A_5^2$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $A_5^2:C_2^4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2$	$G/\operatorname{Fit} \simeq$ $\POPlus(4,5)$
Radical:	$R \simeq$ $C_2^2$	$G/R \simeq$ $\POPlus(4,5)$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times A_5^2$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
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 57600: $A_5^2:C_2^4$

Order 28800: $A_5^2:C_2^3$ x 12, $A_5^2:C_2^3$ x 2, $A_5^2:C_2^3$

Order 14400: $\POPlus(4,5)$ x 16, $A_5^2:C_2^2$ x 12, $A_5^2:C_2^2$ x 6, $C_2^2\times A_5^2$

Order 7200: $\SOPlus(4,4)$ x 8, $\PSOPlus(4,5)$ x 4, $C_2\times A_5^2$ x 3

Order 5760: $C_2^2\times A_4:S_5$

Order 4800: $C_2\times D_{10}.S_5$

Order 3600: $A_5^2$

Order 2880: $C_2\times A_4:S_5$ x 6, $C_2^4:\GL(2,4)$, $\GL(2,4):C_2^4$

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

Order 2304: $C_2^4:D_6^2$

Order 1920: $C_2^4:S_5$

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

Order 1440: $D_6\times S_5$ x 12, $A_4:S_5$ x 4, $C_2^3:\GL(2,4)$ x 3, $\GL(2,4):C_2^3$, $\GL(2,4):C_2^3$, $\GL(2,4):C_2^3$

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

Order 1152: $A_4^2:C_2^3$ x 12, $A_4^2:C_2^3$ x 2, $A_4^2:C_2^3$

Order 960: $C_2^3:S_5$ x 12, $C_2^3\times S_5$ x 3, $C_2^2\times A_4:F_5$, $C_2^4\times A_5$, $C_2^3.S_5$

Order 800: $D_5^2.C_2^3$ x 24, $C_{10}^2:D_4$ x 3, $D_{10}^2.C_2$ x 3, $C_{10}^2:Q_8$

Order 768: $C_2^5:S_4$ x 2

Order 720: $S_3\times S_5$ x 16, $D_6\times A_5$ x 6, $C_6:S_5$ x 6, $C_6\times S_5$ x 6, $C_2^2\times \GL(2,4)$, $A_4\times A_5$

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

Order 576: $\POPlus(4,3)$ x 16, $A_4^2:C_2^2$ x 12, $A_4^2:C_2^2$ x 6, $D_6^2:C_2^2$, $C_2^2\times A_4^2$, $C_2^2:D_6^2$

Order 480: $C_2^2\times S_5$ x 38, $C_2^2:S_5$ x 16, $C_2^3\times A_5$ x 9, $C_2^2.S_5$ x 6, $D_{10}.S_4$ x 6, $C_2\times A_4\times D_{10}$, $C_2\times D_6\times F_5$

Order 400: $D_5^2.C_2^2$ x 64, $D_5^2:C_2^2$ x 36, $D_{10}:F_5$ x 36, $C_2\times C_5^2:Q_8$ x 12, $D_{10}^2$ x 3, $C_{10}^2:C_4$ x 3, $C_{10}^2:C_4$

Order 384: $C_2^4:S_4$ x 24, $C_2^2\wr S_3$ x 4, $C_2^5:A_4$ x 2, $C_2^2\times \GL(2,\mathbb{Z}/4)$

Order 360: $S_3\times A_5$ x 4, $C_3:S_5$ x 4, $C_3\times S_5$ x 4, $C_6\times A_5$ x 3

Order 320: $C_2^4:F_5$

Order 300: $C_5\times A_5$

Order 288: $C_6^2:D_4$ x 28, $D_6\times S_4$ x 12, $A_4\wr C_2$ x 8, $\PSOPlus(4,3)$ x 4, $C_2\times A_4^2$ x 3, $C_2\times D_6^2$ x 2, $C_2\times C_6^2:C_4$, $C_2\times A_4\times D_6$, $C_2\times C_6:S_4$, $C_2\times C_6\times S_4$

Order 256: $C_2^5:D_4$

Order 240: $C_2\times S_5$ x 72, $C_2^2\times A_5$ x 26, $D_6\times F_5$ x 12, $A_4\times D_{10}$ x 6, $A_5:C_4$ x 4, $A_4:F_5$ x 4, $C_2^3:C_{30}$, $D_6\times D_{10}$, $D_{10}.D_6$, $D_{10}:C_{12}$

Order 200: $D_5\wr C_2$ x 48, $D_5:F_5$ x 48, $D_5\times D_{10}$ x 27, $C_{10}:F_5$ x 18, $C_5^2:Q_8$ x 16, $C_{10}:F_5$ x 6, $C_{10}\times D_{10}$ x 3, $C_{10}:D_{10}$

Order 192: $C_2^3:S_4$ x 32, $C_2^3:S_4$ x 24, $C_2\times \GL(2,\mathbb{Z}/4)$ x 12, $C_2^4:A_4$ x 12, $C_2^3\times S_4$ x 3, $C_2^2\wr C_3$ x 2, $A_4\times C_2^4$, $C_{12}:C_2^4$, $C_2^3.S_4$

Order 180: $\GL(2,4)$

Order 160: $C_2^3:F_5$ x 12, $C_2^3\times F_5$ x 4, $C_2^3\times D_{10}$

Order 144: $S_3^2:C_2^2$ x 112, $D_6^2$ x 42, $S_3\times S_4$ x 16, $C_6^2:C_4$ x 14, $A_4\times D_6$ x 6, $C_6:S_4$ x 6, $C_6\times S_4$ x 6, $C_6^2:C_2^2$ x 2, $C_6^2:C_2^2$, $C_2^2:C_6^2$, $A_4^2$

Order 128: $C_2^4:D_4$ x 24, $C_2^4:D_4$ x 3, $C_2^5:C_4$ x 3, $D_4:C_2^4$

Order 120: $C_2\times A_5$ x 21, $S_5$ x 20, $S_3\times F_5$ x 16, $S_3\times D_{10}$ x 12, $C_{30}:C_4$ x 6, $C_6\times F_5$ x 6, $D_5\times A_4$ x 4, $C_{10}\times A_4$ x 3, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$

Order 100: $D_5^2$ x 30, $C_5\times D_{10}$ x 18, $C_5:F_5$ x 12, $C_5:D_{10}$ x 6, $C_5:F_5$ x 4, $C_{10}^2$

Order 96: $C_2^2\times S_4$ x 39, $D_4\times D_6$ x 24, $C_2^2:S_4$ x 16, $\GL(2,\mathbb{Z}/4)$ x 16, $C_2^3\times A_4$ x 9, $C_2^3:A_4$ x 8, $C_2^3:A_4$ x 6, $C_2^2.S_4$ x 6, $C_2^3\times D_6$ x 3, $C_2^3:D_6$ x 2, $C_{12}:C_2^3$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$

Order 80: $C_2^2\times F_5$ x 48, $C_2^2\times D_{10}$ x 16, $D_{10}:C_4$ x 16, $C_2^3\times C_{10}$

Order 72: $S_3\times D_6$ x 140, $\SOPlus(4,2)$ x 64, $C_6\times D_6$ x 28, $C_2\times C_3^2:C_4$ x 28, $C_6:D_6$ x 14, $S_3\times A_4$ x 4, $C_3:S_4$ x 4, $C_3\times S_4$ x 4, $C_6\times A_4$ x 3, $C_2\times C_6^2$

Order 64: $C_2\wr C_2^2$ x 64, $C_2^3:D_4$ x 42, $C_2^4:C_4$ x 36, $D_4:C_2^3$ x 30, $D_4\times C_2^3$ x 7, $C_2^4:C_4$ x 4, $D_4:C_2^3$ x 3, $C_2^6$

Order 60: $S_3\times D_5$ x 16, $D_{30}$ x 6, $S_3\times C_{10}$ x 6, $C_3\times D_{10}$ x 6, $C_{15}:C_4$ x 4, $C_3\times F_5$ x 4, $A_5$ x 3, $C_5\times A_4$, $C_2\times C_{30}$

Order 50: $C_5\times D_5$ x 12, $C_5:D_5$ x 4, $C_5\times C_{10}$ x 3

Order 48: $C_2\times S_4$ x 78, $C_2^2\times D_6$ x 65, $S_3\times D_4$ x 64, $C_2^2\times A_4$ x 27, $C_6:D_4$ x 24, $C_6\times D_4$ x 12, $C_2\times D_{12}$ x 12, $C_4\times D_6$ x 12, $A_4:C_4$ x 4, $C_2^3\times C_6$ x 3, $C_2^2:A_4$ x 2, $C_2^2\times C_{12}$, $C_6.C_2^3$

Order 40: $C_2\times F_5$ x 88, $C_2\times D_{10}$ x 71, $C_2^2\times C_{10}$ x 9

Order 36: $S_3^2$ x 72, $C_6\times S_3$ x 56, $C_6:S_3$ x 28, $C_3^2:C_4$ x 8, $C_6^2$ x 7, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 155, $D_4:C_2^2$ x 136, $C_2^2\wr C_2$ x 80, $D_4:C_2^2$ x 72, $C_2^3:C_4$ x 48, $C_2^3:C_4$ x 39, $C_2^5$ x 15, $C_2^3\times C_4$ x 7, $C_2^2\times Q_8$

Order 30: $D_{15}$ x 4, $C_3\times D_5$ x 4, $C_5\times S_3$ x 4, $C_{30}$ x 3

Order 25: $C_5^2$

Order 24: $C_2\times D_6$ x 257, $C_3:D_4$ x 32, $C_2^2\times C_6$ x 32, $C_2\times A_4$ x 24, $S_4$ x 24, $D_{12}$ x 16, $C_4\times S_3$ x 16, $C_3\times D_4$ x 16, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6

Order 20: $D_{10}$ x 94, $C_2\times C_{10}$ x 26, $F_5$ x 24

Order 18: $C_3\times S_3$ x 16, $C_3:S_3$ x 8, $C_3\times C_6$ x 7

Order 16: $C_2\times D_4$ x 484, $D_4:C_2$ x 192, $C_2^4$ x 98, $C_2^2\times C_4$ x 76, $C_2^2:C_4$ x 46, $C_2\times Q_8$ x 12

Order 15: $C_{15}$

Order 12: $D_6$ x 206, $C_2\times C_6$ x 63, $A_4$ x 4, $C_{12}$ x 4, $C_3:C_4$ x 4

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

Order 9: $C_3^2$

Order 8: $C_2^3$ x 206, $D_4$ x 176, $C_2\times C_4$ x 112, $Q_8$ x 16

Order 6: $S_3$ x 28, $C_6$ x 26

Order 5: $C_5$ x 3

Order 4: $C_2^2$ x 122, $C_4$ x 16

Order 3: $C_3$ x 2

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 57600: $A_5^2:C_2^4$

Order 28800: $A_5^2:C_2^3$, $A_5^2:C_2^3$, $A_5^2:C_2^3$

Order 14400: $C_2^2\times A_5^2$, $A_5^2:C_2^2$, $\POPlus(4,5)$, $A_5^2:C_2^2$

Order 7200: $\PSOPlus(4,5)$, $C_2\times A_5^2$, $\SOPlus(4,4)$

Order 5760: $C_2^2\times A_4:S_5$

Order 4800: $C_2\times D_{10}.S_5$

Order 3600: $A_5^2$

Order 2880: $C_2^4:\GL(2,4)$, $C_2\times A_4:S_5$, $\GL(2,4):C_2^4$

Order 2400: $C_2\times D_{10}\times A_5$, $D_{10}.S_5$

Order 2304: $C_2^4:D_6^2$

Order 1920: $C_2^4:S_5$

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

Order 1440: $D_6\times S_5$ x 2, $\GL(2,4):C_2^3$, $\GL(2,4):C_2^3$, $\GL(2,4):C_2^3$, $C_2^3:\GL(2,4)$, $A_4:S_5$

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

Order 1152: $A_4^2:C_2^3$, $A_4^2:C_2^3$, $A_4^2:C_2^3$

Order 960: $C_2^3\times S_5$ x 2, $C_2^3:S_5$ x 2, $C_2^2\times A_4:F_5$, $C_2^4\times A_5$, $C_2^3.S_5$

Order 800: $C_{10}^2:D_4$ x 2, $D_{10}^2.C_2$ x 2, $D_5^2.C_2^3$ x 2, $C_{10}^2:Q_8$

Order 768: $C_2^5:S_4$

Order 720: $D_6\times A_5$ x 2, $S_3\times S_5$ x 2, $C_2^2\times \GL(2,4)$, $C_6:S_5$, $C_6\times S_5$, $A_4\times A_5$

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

Order 576: $D_6^2:C_2^2$, $C_2^2\times A_4^2$, $C_2^2:D_6^2$, $A_4^2:C_2^2$, $A_4^2:C_2^2$, $\POPlus(4,3)$

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

Order 400: $D_5^2:C_2^2$ x 3, $D_{10}:F_5$ x 3, $D_{10}^2$ x 2, $C_{10}^2:C_4$ x 2, $D_5^2.C_2^2$ x 2, $C_{10}^2:C_4$, $C_2\times C_5^2:Q_8$

Order 384: $C_2^2\wr S_3$ x 2, $C_2^2\times \GL(2,\mathbb{Z}/4)$, $C_2^5:A_4$, $C_2^4:S_4$

Order 360: $S_3\times A_5$ x 2, $C_6\times A_5$, $C_3:S_5$, $C_3\times S_5$

Order 320: $C_2^4:F_5$

Order 300: $C_5\times A_5$

Order 288: $C_6^2:D_4$ x 4, $C_2\times D_6^2$ x 2, $D_6\times S_4$ x 2, $C_2\times C_6^2:C_4$, $C_2\times A_4\times D_6$, $C_2\times C_6:S_4$, $C_2\times C_6\times S_4$, $C_2\times A_4^2$, $\PSOPlus(4,3)$, $A_4\wr C_2$

Order 256: $C_2^5:D_4$

Order 240: $C_2^2\times A_5$ x 8, $C_2\times S_5$ x 6, $A_4\times D_{10}$ x 2, $D_6\times F_5$ x 2, $A_5:C_4$, $C_2^3:C_{30}$, $D_6\times D_{10}$, $D_{10}.D_6$, $D_{10}:C_{12}$, $A_4:F_5$

Order 200: $D_5\times D_{10}$ x 5, $D_5\wr C_2$ x 3, $D_5:F_5$ x 3, $C_{10}\times D_{10}$ x 2, $C_{10}:F_5$ x 2, $C_{10}:D_{10}$, $C_{10}:F_5$, $C_5^2:Q_8$

Order 192: $C_2^3:S_4$ x 2, $C_2^3\times S_4$ x 2, $C_2\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^3:S_4$, $C_2^2\wr C_3$, $A_4\times C_2^4$, $C_{12}:C_2^4$, $C_2^3.S_4$, $C_2^4:A_4$

Order 180: $\GL(2,4)$

Order 160: $C_2^3\times F_5$ x 3, $C_2^3:F_5$ x 2, $C_2^3\times D_{10}$

Order 144: $D_6^2$ x 10, $S_3^2:C_2^2$ x 5, $C_6^2:C_2^2$ x 2, $C_6^2:C_4$ x 2, $A_4\times D_6$ x 2, $S_3\times S_4$ x 2, $C_6^2:C_2^2$, $C_2^2:C_6^2$, $C_6:S_4$, $C_6\times S_4$, $A_4^2$

Order 128: $C_2^4:D_4$ x 2, $C_2^4:D_4$ x 2, $C_2^5:C_4$ x 2, $D_4:C_2^4$

Order 120: $C_2\times A_5$ x 5, $S_3\times D_{10}$ x 4, $S_5$ x 3, $D_5\times A_4$ x 2, $S_3\times F_5$ x 2, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$, $C_{10}\times A_4$, $C_{30}:C_4$, $C_6\times F_5$

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

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

Order 80: $C_2^2\times F_5$ x 9, $C_2^2\times D_{10}$ x 7, $D_{10}:C_4$ x 2, $C_2^3\times C_{10}$

Order 72: $S_3\times D_6$ x 16, $C_6\times D_6$ x 6, $C_6:D_6$ x 4, $\SOPlus(4,2)$ x 3, $C_2\times C_3^2:C_4$ x 2, $S_3\times A_4$ x 2, $C_2\times C_6^2$, $C_6\times A_4$, $C_3:S_4$, $C_3\times S_4$

Order 64: $D_4\times C_2^3$ x 5, $C_2^3:D_4$ x 5, $D_4:C_2^3$ x 3, $C_2^4:C_4$ x 3, $C_2^4:C_4$ x 2, $D_4:C_2^3$ x 2, $C_2\wr C_2^2$ x 2, $C_2^6$

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

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

Order 48: $C_2^2\times D_6$ x 19, $C_2^2\times A_4$ x 9, $C_2\times S_4$ x 7, $S_3\times D_4$ x 5, $C_6:D_4$ x 4, $C_2^3\times C_6$ x 3, $C_6\times D_4$ x 2, $C_2\times D_{12}$ x 2, $C_4\times D_6$ x 2, $C_2^2:A_4$, $C_2^2\times C_{12}$, $C_6.C_2^3$, $A_4:C_4$

Order 40: $C_2\times D_{10}$ x 20, $C_2\times F_5$ x 9, $C_2^2\times C_{10}$ x 4

Order 36: $S_3^2$ x 11, $C_6\times S_3$ x 7, $C_6:S_3$ x 5, $C_6^2$ x 2, $C_3^2:C_4$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 22, $C_2^5$ x 8, $C_2^2\wr C_2$ x 7, $C_2^3:C_4$ x 6, $C_2^3\times C_4$ x 5, $D_4:C_2^2$ x 4, $D_4:C_2^2$ x 4, $C_2^3:C_4$ x 2, $C_2^2\times Q_8$

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

Order 25: $C_5^2$

Order 24: $C_2\times D_6$ x 39, $C_2^2\times C_6$ x 9, $C_2\times A_4$ x 6, $C_3:D_4$ x 4, $S_4$ x 4, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_3\times D_4$ x 2, $C_2\times C_{12}$, $C_6:C_4$

Order 20: $D_{10}$ x 18, $C_2\times C_{10}$ x 8, $F_5$ x 4

Order 18: $C_3\times S_3$ x 4, $C_3:S_3$ x 3, $C_3\times C_6$ x 2

Order 16: $C_2^4$ x 30, $C_2\times D_4$ x 30, $C_2^2\times C_4$ x 11, $C_2^2:C_4$ x 6, $D_4:C_2$ x 4, $C_2\times Q_8$

Order 15: $C_{15}$

Order 12: $D_6$ x 30, $C_2\times C_6$ x 13, $A_4$ x 3, $C_{12}$, $C_3:C_4$

Order 10: $D_5$ x 7, $C_{10}$ x 5

Order 9: $C_3^2$

Order 8: $C_2^3$ x 44, $D_4$ x 12, $C_2\times C_4$ x 11, $Q_8$

Order 6: $S_3$ x 9, $C_6$ x 7

Order 5: $C_5$ x 2

Order 4: $C_2^2$ x 28, $C_4$ x 3

Order 3: $C_3$ x 2

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 57600: $A_5^2:C_2^4$ ($C_1$)

Order 28800: $A_5^2:C_2^3$ ($C_2$) x 12, $A_5^2:C_2^3$ ($C_2$) x 2, $A_5^2:C_2^3$ ($C_2$)

Order 14400: $\POPlus(4,5)$ ($C_2^2$) x 16, $A_5^2:C_2^2$ ($C_2^2$) x 12, $A_5^2:C_2^2$ ($C_2^2$) x 6, $C_2^2\times A_5^2$ ($C_2^2$)

Order 7200: $\SOPlus(4,4)$ ($C_2^3$) x 8, $\PSOPlus(4,5)$ ($C_2^3$) x 4, $C_2\times A_5^2$ ($C_2^3$) x 3

Order 3600: $A_5^2$ ($C_2^4$)

Order 4: $C_2^2$ ($\POPlus(4,5)$)

Order 2: $C_2$ ($A_5^2:C_2^3$) x 3

Order 1: $C_1$ ($A_5^2:C_2^4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 57600: $A_5^2:C_2^4$ ($C_1$)

Order 28800: $A_5^2:C_2^3$ ($C_2$), $A_5^2:C_2^3$ ($C_2$), $A_5^2:C_2^3$ ($C_2$)

Order 14400: $C_2^2\times A_5^2$ ($C_2^2$), $A_5^2:C_2^2$ ($C_2^2$), $\POPlus(4,5)$ ($C_2^2$), $A_5^2:C_2^2$ ($C_2^2$)

Order 7200: $\PSOPlus(4,5)$ ($C_2^3$), $C_2\times A_5^2$ ($C_2^3$), $\SOPlus(4,4)$ ($C_2^3$)

Order 3600: $A_5^2$ ($C_2^4$)

Order 4: $C_2^2$ ($\POPlus(4,5)$)

Order 2: $C_2$ ($A_5^2:C_2^3$)

Order 1: $C_1$ ($A_5^2:C_2^4$)

Series

Derived series

$A_5^2:C_2^4$

$\rhd$

$A_5^2:C_2^4$

$\rhd$

$A_5^2$

$\rhd$

$A_5^2$

Chief series

$A_5^2:C_2^4$

$\rhd$

$A_5^2:C_2^4$

$\rhd$

$A_5^2:C_2^3$

$\rhd$

$A_5^2:C_2^3$

$\rhd$

$C_2^2\times A_5^2$

$\rhd$

$C_2^2\times A_5^2$

$\rhd$

$C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$A_5^2:C_2^4$

$\rhd$

$A_5^2:C_2^4$

$\rhd$

$A_5^2$

$\rhd$

$A_5^2$

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 4 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $100 \times 100$ rational character table. Alternatively, you may search for characters of this group with desired properties.