Abstract group 8640.bm: $A_5:D_6^2$

• Label: 8640.bm
• Order: \( 2^{6} \cdot 3^{3} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{3} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $13$
• Trans deg.: $60$
• Rank: $4$

Group information

Description:	$A_5:D_6^2$
Order:	\(8640\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$D_6\wr C_2.C_2.S_5$, of order \(69120\)\(\medspace = 2^{9} \cdot 3^{3} \cdot 5 \)
Composition factors:	$C_2$ x 4, $C_3$ x 2, $A_5$
Derived length:	$2$

This group is nonabelian, nonsolvable, and rational.

Group statistics

Order	1	2	3	4	5	6	10	12	15	30
Elements	1	831	188	960	24	3732	744	1200	192	768	8640
Conjugacy classes	1	23	7	8	1	59	7	10	3	7	126
Divisions	1	23	7	8	1	59	7	10	3	7	126
Autjugacy classes	1	10	5	3	1	23	3	3	2	3	54

Dimension	1	2	4	5	6	8	10	12	16	20	24
Irr. complex chars.	16	16	20	16	8	16	16	8	4	4	2	126
Irr. rational chars.	16	16	20	16	8	16	16	8	4	4	2	126

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$60$
Rank:	$4$
Inequivalent generating quadruples:	$22688544060$

Minimal degrees of faithful linear representations

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

Constructions

Permutation group:	Degree $13$ $\langle(9,10), (1,2)(3,4,5)(6,7)(11,12), (2,3)(7,8)(9,10)(11,12,13), (6,7,8)(11,13,12), (6,8,7)(11,13,12), (6,7)(9,10)(11,13,12), (6,8)(11,12)\rangle$
Direct product:	$C_2$ $\, \times\, $ $S_3$ ${}^2$ $\, \times\, $ $S_5$
Semidirect product:	$A_5$ $\,\rtimes\,$ $D_6^2$	$(C_6\times S_5)$ $\,\rtimes\,$ $D_6$	$(C_6:S_5)$ $\,\rtimes\,$ $D_6$	$C_6$ $\,\rtimes\,$ $(D_6\times S_5)$	all 40
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(\GL(2,4):S_3^2)$

Elements of the group are displayed as permutations of degree 13.

Homology

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

Subgroups

There are 234496 subgroups in 5371 conjugacy classes, 132 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_3^2\times S_5$
Commutator:	$G' \simeq$ $C_3\times \GL(2,4)$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $A_5:D_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6$	$G/\operatorname{Fit} \simeq$ $C_2^2\times S_5$
Radical:	$R \simeq$ $S_3\times D_6$	$G/R \simeq$ $S_5$
Socle:	$\operatorname{soc} \simeq$ $C_6\times \GL(2,4)$	$G/\operatorname{soc} \simeq$ $C_2^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

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 8640: $A_5:D_6^2$

Order 4320: $S_3^2\times S_5$ x 8, $S_3\times C_6:S_5$ x 2, $C_6\times S_3\times S_5$ x 2, $S_3\times D_6\times A_5$, $C_2\times A_5:S_3^2$, $C_6:S_3\times S_5$

Order 2880: $\GL(2,4):C_2^4$ x 2

Order 2160: $A_5:S_3^2$ x 8, $\GL(2,4):D_6$ x 8, $\GL(2,4):D_6$ x 4, $S_3^2\times A_5$ x 4, $A_5:S_3^2$ x 4, $\GL(2,4):D_6$ x 2, $D_6\times \GL(2,4)$ x 2, $A_5:C_6^2$, $\GL(2,4):D_6$, $\GL(2,4):D_6$

Order 1728: $A_4:D_6^2$

Order 1440: $D_6\times S_5$ x 25, $\GL(2,4):C_2^3$ x 2, $\GL(2,4):C_2^3$ x 2, $\GL(2,4):C_2^3$ x 2, $D_5.D_6^2$

Order 1080: $S_3\times \GL(2,4)$ x 4, $C_3^2:S_5$ x 4, $\GL(2,4):S_3$ x 2, $C_3^2\times S_5$ x 2, $C_3^2:S_5$ x 2, $C_6\times \GL(2,4)$

Order 960: $C_2^3\times S_5$

Order 864: $S_4\times S_3^2$ x 8, $S_3\times C_6:S_4$ x 2, $C_6\times S_3\times S_4$ x 2, $S_3\times D_6^2$, $D_6^2:C_6$, $C_2\times A_4:S_3^2$, $C_6:S_3\times S_4$

Order 720: $S_3\times S_5$ x 36, $D_6\times A_5$ x 13, $C_6:S_5$ x 13, $C_6\times S_5$ x 13, $F_5\times S_3^2$ x 8, $D_{10}.S_3^2$ x 2, $D_{30}:C_{12}$ x 2, $C_2^2\times \GL(2,4)$ x 2, $D_{10}\times S_3^2$, $D_{10}.S_3^2$, $C_6:S_3\times F_5$

Order 576: $C_2^2:D_6^2$ x 2, $C_4:D_6^2$

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

Order 480: $C_2^2\times S_5$ x 14, $C_2\times D_6\times F_5$ x 2, $C_2^3\times A_5$

Order 432: $C_2\times S_3^3$ x 24, $A_4:S_3^2$ x 8, $C_6^2:D_6$ x 8, $A_4\times S_3^2$ x 4, $A_4:S_3^2$ x 4, $C_6^2:D_6$ x 4, $C_3:D_6^2$ x 3, $C_3\times D_6^2$ x 3, $C_6\times S_3\times A_4$ x 2, $C_6^2:D_6$ x 2, $C_3:D_6^2$, $C_6:S_3\times A_4$, $C_6^2:D_6$, $A_4:C_6^2$

Order 360: $S_3\times A_5$ x 10, $C_3:S_5$ x 10, $C_3\times S_5$ x 10, $D_5\times S_3^2$ x 8, $D_5.S_3^2$ x 8, $D_{15}:C_{12}$ x 8, $C_6\times A_5$ x 7, $D_5.S_3^2$ x 4, $C_3:S_3\times F_5$ x 4, $S_3\times D_{30}$ x 2, $C_{30}:D_6$ x 2, $C_{30}:C_{12}$ x 2, $C_{30}:D_6$, $C_{10}\times S_3^2$, $C_{30}:D_6$, $C_2\times C_3^2:F_5$, $C_{30}:C_{12}$

Order 288: $D_6\times S_4$ x 27, $D_4\times S_3^2$ x 16, $D_6^2:C_2$ x 4, $C_2\times D_6^2$ x 4, $C_6^2:C_2^3$ x 2, $D_6^2:C_2$ x 2, $D_6\times D_{12}$ x 2, $C_2\times A_4\times D_6$ x 2, $C_2\times C_6:S_4$ x 2, $C_2\times C_6\times S_4$ x 2, $D_{12}:D_6$, $C_2.D_6^2$, $C_6^2:C_2^3$

Order 240: $C_2\times S_5$ x 28, $D_6\times F_5$ x 25, $C_2^2\times A_5$ x 7, $D_6\times D_{10}$ x 2, $D_{10}.D_6$ x 2, $D_{10}:C_{12}$ x 2

Order 216: $S_3^3$ x 64, $C_6:S_3^2$ x 36, $C_6\times S_3^2$ x 36, $C_6:S_3^2$ x 12, $C_6^2:C_6$ x 4, $C_3^2:S_4$ x 4, $C_6^2:C_6$ x 3, $S_3\times C_6^2$ x 3, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_3^2\times S_4$ x 2, $C_6^2:S_3$, $C_6^2:C_6$

Order 192: $C_{12}:C_2^4$ x 2, $C_2^3\times S_4$

Order 180: $S_3\times D_{15}$ x 8, $C_{15}:D_6$ x 8, $C_{15}:D_6$ x 4, $C_5\times S_3^2$ x 4, $C_{15}:D_6$ x 4, $C_{15}:C_{12}$ x 4, $\GL(2,4)$ x 3, $C_3\times D_{30}$ x 2, $S_3\times C_{30}$ x 2, $C_3^2:F_5$ x 2, $C_3^2\times F_5$ x 2, $C_3:D_{30}$, $C_{30}:S_3$, $C_3^2\times D_{10}$

Order 160: $C_2^3\times F_5$

Order 144: $D_6^2$ x 90, $S_3\times S_4$ x 44, $D_6:D_6$ x 32, $C_6:S_4$ x 16, $C_{12}:D_6$ x 16, $D_6:D_6$ x 16, $S_3\times D_{12}$ x 16, $A_4\times D_6$ x 15, $C_6\times S_4$ x 15, $C_6^2:C_2^2$ x 8, $C_{12}:D_6$ x 8, $C_{12}:D_6$ x 8, $C_4\times S_3^2$ x 8, $C_6^2:C_2^2$ x 4, $C_6^2:C_2^2$ x 4, $C_6:D_{12}$ x 4, $C_2^2:C_6^2$ x 2, $C_6^2:C_2^2$ x 2, $C_6\times D_{12}$ x 2, $C_{12}\times D_6$ x 2, $D_6:D_6$ x 2, $D_6.D_6$ x 2, $C_4:C_6^2$, $C_6:D_{12}$, $C_{12}:D_6$, $C_2^2.S_3^2$

Order 120: $S_3\times F_5$ x 36, $S_3\times D_{10}$ x 25, $C_{30}:C_4$ x 13, $C_6\times F_5$ x 13, $S_5$ x 8, $C_2\times A_5$ x 7, $C_2\times D_{30}$ x 2, $C_{10}\times D_6$ x 2, $C_6\times D_{10}$ x 2

Order 108: $C_3:S_3^2$ x 48, $C_3\times S_3^2$ x 48, $C_3^2:D_6$ x 18, $C_3^2\times D_6$ x 18, $C_3:S_3^2$ x 16, $C_3^2:D_6$ x 6, $C_3\times C_6^2$, $C_3^2\times A_4$

Order 96: $D_4\times D_6$ x 49, $C_2^2\times S_4$ x 16, $C_2^3\times D_6$ x 5, $C_2^3:D_6$ x 4, $C_{12}:C_2^3$ x 2, $C_2^2\times D_{12}$ x 2, $C_{12}:C_2^3$ x 2, $C_2^3\times A_4$

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, $C_3\times C_{30}$

Order 80: $C_2^2\times F_5$ x 14, $C_2^2\times D_{10}$

Order 72: $S_3\times D_6$ x 347, $C_6\times D_6$ x 92, $C_6:D_6$ x 47, $C_3:S_4$ x 16, $C_6\wr C_2$ x 16, $C_3:D_{12}$ x 16, $S_3\times A_4$ x 14, $C_3\times S_4$ x 14, $C_6\times A_4$ x 10, $C_6^2:C_2$ x 8, $C_3\times D_{12}$ x 8, $S_3\times C_{12}$ x 8, $D_6:S_3$ x 8, $C_6.D_6$ x 8, $C_2\times C_6^2$ x 4, $D_4\times C_3^2$ x 4, $C_3:D_{12}$ x 4, $C_{12}:S_3$ x 4, $C_6.D_6$ x 4, $C_6:C_{12}$ x 2, $C_6\times C_{12}$, $C_6.D_6$

Order 64: $D_4\times C_2^3$

Order 60: $S_3\times D_5$ x 36, $D_{30}$ x 13, $S_3\times C_{10}$ x 13, $C_3\times D_{10}$ x 13, $C_{15}:C_4$ x 10, $C_3\times F_5$ x 10, $C_2\times C_{30}$ x 2, $A_5$

Order 54: $C_3^2:C_6$ x 12, $S_3\times C_3^2$ x 12, $C_3^2:S_3$ x 4, $C_3^2\times C_6$ x 3

Order 48: $S_3\times D_4$ x 136, $C_2^2\times D_6$ x 104, $C_6:D_4$ x 50, $C_2\times S_4$ x 41, $C_6\times D_4$ x 25, $C_2\times D_{12}$ x 25, $C_4\times D_6$ x 25, $C_2^2\times A_4$ x 9, $C_2^3\times C_6$ x 5, $C_2^2\times C_{12}$ x 2, $C_6.C_2^3$ x 2

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

Order 40: $C_2\times F_5$ x 28, $C_2\times D_{10}$ x 14, $C_2^2\times C_{10}$

Order 36: $S_3^2$ x 220, $C_6\times S_3$ x 190, $C_6:S_3$ x 101, $C_6^2$ x 25, $C_3\times A_4$ x 6, $C_3:C_{12}$ x 4, $C_3\times C_{12}$ x 2, $C_3^2:C_4$ x 2

Order 32: $C_2^2\times D_4$ x 28, $C_2^5$ x 2, $C_2^3\times C_4$

Order 30: $D_{15}$ x 10, $C_3\times D_5$ x 10, $C_5\times S_3$ x 10, $C_{30}$ x 7

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 418, $C_3:D_4$ x 72, $C_2^2\times C_6$ x 53, $D_{12}$ x 36, $C_4\times S_3$ x 36, $C_3\times D_4$ x 36, $S_4$ x 18, $C_2\times A_4$ x 14, $C_2\times C_{12}$ x 13, $C_6:C_4$ x 13

Order 20: $D_{10}$ x 28, $F_5$ x 8, $C_2\times C_{10}$ x 7

Order 18: $C_3\times S_3$ x 68, $C_3:S_3$ x 38, $C_3\times C_6$ x 31

Order 16: $C_2\times D_4$ x 112, $C_2^4$ x 37, $C_2^2\times C_4$ x 14

Order 15: $C_{15}$ x 3

Order 12: $D_6$ x 375, $C_2\times C_6$ x 113, $C_{12}$ x 10, $C_3:C_4$ x 10, $A_4$ x 4

Order 10: $D_5$ x 8, $C_{10}$ x 7

Order 9: $C_3^2$ x 7

Order 8: $C_2^3$ x 134, $D_4$ x 64, $C_2\times C_4$ x 28

Order 6: $S_3$ x 66, $C_6$ x 59

Order 5: $C_5$

Order 4: $C_2^2$ x 114, $C_4$ x 8

Order 3: $C_3$ x 7

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 8640: $A_5:D_6^2$

Order 4320: $S_3\times D_6\times A_5$, $S_3\times C_6:S_5$, $C_2\times A_5:S_3^2$, $C_6:S_3\times S_5$, $C_6\times S_3\times S_5$, $S_3^2\times S_5$

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

Order 2160: $A_5:C_6^2$, $\GL(2,4):D_6$, $\GL(2,4):D_6$, $\GL(2,4):D_6$, $D_6\times \GL(2,4)$, $\GL(2,4):D_6$, $S_3^2\times A_5$, $A_5:S_3^2$, $A_5:S_3^2$, $\GL(2,4):D_6$

Order 1728: $A_4:D_6^2$

Order 1440: $D_6\times S_5$ x 6, $D_5.D_6^2$, $\GL(2,4):C_2^3$, $\GL(2,4):C_2^3$, $\GL(2,4):C_2^3$

Order 1080: $C_6\times \GL(2,4)$, $S_3\times \GL(2,4)$, $\GL(2,4):S_3$, $C_3^2\times S_5$, $C_3^2:S_5$, $C_3^2:S_5$

Order 960: $C_2^3\times S_5$

Order 864: $S_3\times D_6^2$, $D_6^2:C_6$, $C_2\times A_4:S_3^2$, $S_3\times C_6:S_4$, $C_6:S_3\times S_4$, $C_6\times S_3\times S_4$, $S_4\times S_3^2$

Order 720: $S_3\times S_5$ x 5, $D_6\times A_5$ x 4, $C_6:S_5$ x 4, $C_6\times S_5$ x 4, $D_{10}\times S_3^2$, $D_{10}.S_3^2$, $D_{10}.S_3^2$, $C_6:S_3\times F_5$, $D_{30}:C_{12}$, $F_5\times S_3^2$, $C_2^2\times \GL(2,4)$

Order 576: $C_2^2:D_6^2$, $C_4:D_6^2$

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

Order 480: $C_2^2\times S_5$ x 5, $C_2\times D_6\times F_5$, $C_2^3\times A_5$

Order 432: $C_2\times S_3^3$ x 8, $C_3:D_6^2$ x 2, $C_3\times D_6^2$ x 2, $C_3:D_6^2$, $C_6:S_3\times A_4$, $C_6\times S_3\times A_4$, $C_6^2:D_6$, $C_6^2:D_6$, $A_4:C_6^2$, $A_4\times S_3^2$, $A_4:S_3^2$, $A_4:S_3^2$, $C_6^2:D_6$, $C_6^2:D_6$

Order 360: $C_6\times A_5$ x 3, $S_3\times A_5$ x 3, $C_3:S_5$ x 3, $C_3\times S_5$ x 3, $D_5\times S_3^2$ x 2, $C_{30}:D_6$, $S_3\times D_{30}$, $C_{10}\times S_3^2$, $C_{30}:D_6$, $C_{30}:D_6$, $C_2\times C_3^2:F_5$, $C_{30}:C_{12}$, $C_{30}:C_{12}$, $D_5.S_3^2$, $D_5.S_3^2$, $C_3:S_3\times F_5$, $D_{15}:C_{12}$

Order 288: $D_6\times S_4$ x 7, $C_2\times D_6^2$ x 3, $D_6^2:C_2$ x 2, $D_6^2:C_2$ x 2, $D_4\times S_3^2$ x 2, $C_6^2:C_2^3$, $D_{12}:D_6$, $D_6\times D_{12}$, $C_2.D_6^2$, $C_2\times A_4\times D_6$, $C_2\times C_6:S_4$, $C_2\times C_6\times S_4$, $C_6^2:C_2^3$

Order 240: $C_2\times S_5$ x 7, $D_6\times F_5$ x 6, $C_2^2\times A_5$ x 3, $D_6\times D_{10}$, $D_{10}.D_6$, $D_{10}:C_{12}$

Order 216: $C_6:S_3^2$ x 11, $C_6\times S_3^2$ x 10, $S_3^3$ x 9, $C_6:S_3^2$ x 4, $C_6^2:C_6$ x 2, $S_3\times C_6^2$ x 2, $C_6^2:S_3$, $C_6^2:C_6$, $C_6^2:C_6$, $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2:S_4$, $C_3^2\times S_4$

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

Order 180: $C_{15}:D_6$ x 2, $C_{15}:D_6$ x 2, $\GL(2,4)$ x 2, $C_3:D_{30}$, $C_{30}:S_3$, $C_3\times D_{30}$, $S_3\times C_{30}$, $C_3^2\times D_{10}$, $C_{15}:D_6$, $S_3\times D_{15}$, $C_5\times S_3^2$, $C_3^2:F_5$, $C_{15}:C_{12}$, $C_3^2\times F_5$

Order 160: $C_2^3\times F_5$

Order 144: $D_6^2$ x 28, $C_6:S_4$ x 6, $S_3\times S_4$ x 6, $A_4\times D_6$ x 5, $C_6\times S_4$ x 5, $C_6^2:C_2^2$ x 4, $C_6^2:C_2^2$ x 3, $D_6:D_6$ x 3, $D_6:D_6$ x 3, $C_6^2:C_2^2$ x 2, $C_{12}:D_6$ x 2, $C_6^2:C_2^2$ x 2, $C_{12}:D_6$ x 2, $C_6:D_{12}$ x 2, $C_2^2:C_6^2$, $C_4:C_6^2$, $C_6:D_{12}$, $C_{12}:D_6$, $C_6\times D_{12}$, $C_{12}\times D_6$, $D_6:D_6$, $C_2^2.S_3^2$, $D_6.D_6$, $C_{12}:D_6$, $S_3\times D_{12}$, $C_4\times S_3^2$

Order 120: $S_3\times D_{10}$ x 7, $S_3\times F_5$ x 5, $C_{30}:C_4$ x 4, $C_6\times F_5$ x 4, $C_2\times A_5$ x 3, $S_5$ x 3, $C_2\times D_{30}$, $C_{10}\times D_6$, $C_6\times D_{10}$

Order 108: $C_3:S_3^2$ x 10, $C_3\times S_3^2$ x 9, $C_3^2\times D_6$ x 7, $C_3^2:D_6$ x 6, $C_3^2:D_6$ x 3, $C_3:S_3^2$ x 3, $C_3\times C_6^2$, $C_3^2\times A_4$

Order 96: $D_4\times D_6$ x 11, $C_2^2\times S_4$ x 6, $C_2^3\times D_6$ x 3, $C_2^3:D_6$ x 2, $C_2^3\times A_4$, $C_{12}:C_2^3$, $C_2^2\times D_{12}$, $C_{12}:C_2^3$

Order 90: $C_3^2\times D_5$ x 2, $C_3:D_{15}$, $C_{15}:S_3$, $C_3\times D_{15}$, $S_3\times C_{15}$, $C_3\times C_{30}$

Order 80: $C_2^2\times F_5$ x 5, $C_2^2\times D_{10}$

Order 72: $S_3\times D_6$ x 66, $C_6\times D_6$ x 28, $C_6:D_6$ x 19, $C_6\times A_4$ x 5, $C_3:S_4$ x 5, $S_3\times A_4$ x 4, $C_3\times S_4$ x 4, $C_2\times C_6^2$ x 3, $C_6^2:C_2$ x 3, $C_6\wr C_2$ x 3, $D_4\times C_3^2$ x 2, $C_3:D_{12}$ x 2, $C_6\times C_{12}$, $C_6.D_6$, $C_3:D_{12}$, $C_{12}:S_3$, $C_6:C_{12}$, $C_3\times D_{12}$, $S_3\times C_{12}$, $D_6:S_3$, $C_6.D_6$, $C_6.D_6$

Order 64: $D_4\times C_2^3$

Order 60: $S_3\times D_5$ x 8, $C_3\times D_{10}$ x 5, $D_{30}$ x 4, $S_3\times C_{10}$ x 4, $C_{15}:C_4$ x 3, $C_3\times F_5$ x 3, $A_5$, $C_2\times C_{30}$

Order 54: $S_3\times C_3^2$ x 5, $C_3^2:C_6$ x 4, $C_3^2\times C_6$ x 2, $C_3^2:S_3$ x 2

Order 48: $C_2^2\times D_6$ x 34, $S_3\times D_4$ x 14, $C_6:D_4$ x 13, $C_2\times S_4$ x 11, $C_6\times D_4$ x 7, $C_2\times D_{12}$ x 6, $C_4\times D_6$ x 6, $C_2^2\times A_4$ 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: $C_2\times F_5$ x 7, $C_2\times D_{10}$ x 6, $C_2^2\times C_{10}$

Order 36: $C_6\times S_3$ x 47, $S_3^2$ x 38, $C_6:S_3$ x 30, $C_6^2$ x 13, $C_3\times A_4$ x 4, $C_3\times C_{12}$, $C_3^2:C_4$, $C_3:C_{12}$

Order 32: $C_2^2\times D_4$ x 10, $C_2^5$ x 2, $C_2^3\times C_4$

Order 30: $C_3\times D_5$ x 5, $C_{30}$ x 3, $D_{15}$ x 3, $C_5\times S_3$ x 3

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 97, $C_2^2\times C_6$ x 20, $C_3:D_4$ x 13, $C_3\times D_4$ x 8, $C_2\times A_4$ x 6, $S_4$ x 6, $D_{12}$ x 5, $C_4\times S_3$ x 5, $C_2\times C_{12}$ x 4, $C_6:C_4$ x 4

Order 20: $D_{10}$ x 9, $C_2\times C_{10}$ x 3, $F_5$ x 3

Order 18: $C_3\times S_3$ x 21, $C_3\times C_6$ x 14, $C_3:S_3$ x 13

Order 16: $C_2\times D_4$ x 22, $C_2^4$ x 16, $C_2^2\times C_4$ x 5

Order 15: $C_{15}$ x 2

Order 12: $D_6$ x 86, $C_2\times C_6$ x 38, $A_4$ x 3, $C_{12}$ x 3, $C_3:C_4$ x 3

Order 10: $D_5$ x 4, $C_{10}$ x 3

Order 9: $C_3^2$ x 5

Order 8: $C_2^3$ x 41, $D_4$ x 13, $C_2\times C_4$ x 7

Order 6: $C_6$ x 23, $S_3$ x 22

Order 5: $C_5$

Order 4: $C_2^2$ x 36, $C_4$ x 3

Order 3: $C_3$ x 5

Order 2: $C_2$ x 10

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 8640: $A_5:D_6^2$ ($C_1$)

Order 4320: $S_3^2\times S_5$ ($C_2$) x 8, $S_3\times C_6:S_5$ ($C_2$) x 2, $C_6\times S_3\times S_5$ ($C_2$) x 2, $S_3\times D_6\times A_5$ ($C_2$), $C_2\times A_5:S_3^2$ ($C_2$), $C_6:S_3\times S_5$ ($C_2$)

Order 2160: $A_5:S_3^2$ ($C_2^2$) x 8, $\GL(2,4):D_6$ ($C_2^2$) x 8, $\GL(2,4):D_6$ ($C_2^2$) x 4, $S_3^2\times A_5$ ($C_2^2$) x 4, $A_5:S_3^2$ ($C_2^2$) x 4, $\GL(2,4):D_6$ ($C_2^2$) x 2, $D_6\times \GL(2,4)$ ($C_2^2$) x 2, $A_5:C_6^2$ ($C_2^2$), $\GL(2,4):D_6$ ($C_2^2$), $\GL(2,4):D_6$ ($C_2^2$)

Order 1440: $D_6\times S_5$ ($S_3$) x 2

Order 1080: $S_3\times \GL(2,4)$ ($C_2^3$) x 4, $C_3^2:S_5$ ($C_2^3$) x 4, $\GL(2,4):S_3$ ($C_2^3$) x 2, $C_3^2\times S_5$ ($C_2^3$) x 2, $C_3^2:S_5$ ($C_2^3$) x 2, $C_6\times \GL(2,4)$ ($C_2^3$)

Order 720: $S_3\times S_5$ ($D_6$) x 8, $D_6\times A_5$ ($D_6$) x 2, $C_6:S_5$ ($D_6$) x 2, $C_6\times S_5$ ($D_6$) x 2

Order 540: $C_3\times \GL(2,4)$ ($C_2^4$)

Order 360: $S_3\times A_5$ ($C_2\times D_6$) x 4, $C_3:S_5$ ($C_2\times D_6$) x 4, $C_3\times S_5$ ($C_2\times D_6$) x 4, $C_6\times A_5$ ($C_2\times D_6$) x 2

Order 240: $C_2\times S_5$ ($S_3^2$)

Order 180: $\GL(2,4)$ ($C_2^2\times D_6$) x 2

Order 120: $S_5$ ($S_3\times D_6$) x 2, $C_2\times A_5$ ($S_3\times D_6$)

Order 72: $S_3\times D_6$ ($S_5$)

Order 60: $A_5$ ($D_6^2$)

Order 36: $S_3^2$ ($C_2\times S_5$) x 4, $C_6\times S_3$ ($C_2\times S_5$) x 2, $C_6:S_3$ ($C_2\times S_5$)

Order 18: $C_3\times S_3$ ($C_2^2\times S_5$) x 4, $C_3:S_3$ ($C_2^2\times S_5$) x 2, $C_3\times C_6$ ($C_2^2\times S_5$)

Order 12: $D_6$ ($S_3\times S_5$) x 2

Order 9: $C_3^2$ ($C_2^3\times S_5$)

Order 6: $S_3$ ($D_6\times S_5$) x 4, $C_6$ ($D_6\times S_5$) x 2

Order 3: $C_3$ ($\GL(2,4):C_2^4$) x 2

Order 2: $C_2$ ($S_3^2\times S_5$)

Order 1: $C_1$ ($A_5:D_6^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 8640: $A_5:D_6^2$ ($C_1$)

Order 4320: $S_3\times D_6\times A_5$ ($C_2$), $S_3\times C_6:S_5$ ($C_2$), $C_2\times A_5:S_3^2$ ($C_2$), $C_6:S_3\times S_5$ ($C_2$), $C_6\times S_3\times S_5$ ($C_2$), $S_3^2\times S_5$ ($C_2$)

Order 2160: $A_5:C_6^2$ ($C_2^2$), $\GL(2,4):D_6$ ($C_2^2$), $\GL(2,4):D_6$ ($C_2^2$), $\GL(2,4):D_6$ ($C_2^2$), $D_6\times \GL(2,4)$ ($C_2^2$), $\GL(2,4):D_6$ ($C_2^2$), $S_3^2\times A_5$ ($C_2^2$), $A_5:S_3^2$ ($C_2^2$), $A_5:S_3^2$ ($C_2^2$), $\GL(2,4):D_6$ ($C_2^2$)

Order 1440: $D_6\times S_5$ ($S_3$)

Order 1080: $C_6\times \GL(2,4)$ ($C_2^3$), $S_3\times \GL(2,4)$ ($C_2^3$), $\GL(2,4):S_3$ ($C_2^3$), $C_3^2\times S_5$ ($C_2^3$), $C_3^2:S_5$ ($C_2^3$), $C_3^2:S_5$ ($C_2^3$)

Order 720: $D_6\times A_5$ ($D_6$), $C_6:S_5$ ($D_6$), $C_6\times S_5$ ($D_6$), $S_3\times S_5$ ($D_6$)

Order 540: $C_3\times \GL(2,4)$ ($C_2^4$)

Order 360: $C_6\times A_5$ ($C_2\times D_6$), $S_3\times A_5$ ($C_2\times D_6$), $C_3:S_5$ ($C_2\times D_6$), $C_3\times S_5$ ($C_2\times D_6$)

Order 240: $C_2\times S_5$ ($S_3^2$)

Order 180: $\GL(2,4)$ ($C_2^2\times D_6$)

Order 120: $C_2\times A_5$ ($S_3\times D_6$), $S_5$ ($S_3\times D_6$)

Order 72: $S_3\times D_6$ ($S_5$)

Order 60: $A_5$ ($D_6^2$)

Order 36: $C_6:S_3$ ($C_2\times S_5$), $C_6\times S_3$ ($C_2\times S_5$), $S_3^2$ ($C_2\times S_5$)

Order 18: $C_3\times C_6$ ($C_2^2\times S_5$), $C_3:S_3$ ($C_2^2\times S_5$), $C_3\times S_3$ ($C_2^2\times S_5$)

Order 12: $D_6$ ($S_3\times S_5$)

Order 9: $C_3^2$ ($C_2^3\times S_5$)

Order 6: $C_6$ ($D_6\times S_5$), $S_3$ ($D_6\times S_5$)

Order 3: $C_3$ ($\GL(2,4):C_2^4$)

Order 2: $C_2$ ($S_3^2\times S_5$)

Order 1: $C_1$ ($A_5:D_6^2$)

Series

Derived series

$A_5:D_6^2$

$\rhd$

$A_5:D_6^2$

$\rhd$

$C_3\times \GL(2,4)$

$\rhd$

$C_3\times \GL(2,4)$

$\rhd$

$A_5$

$\rhd$

$A_5$

Chief series

$A_5:D_6^2$

$\rhd$

$A_5:D_6^2$

$\rhd$

$S_3\times D_6\times A_5$

$\rhd$

$S_3\times D_6\times A_5$

$\rhd$

$S_3\times D_6$

$\rhd$

$S_3\times D_6$

$\rhd$

$C_6\times S_3$

$\rhd$

$C_6\times S_3$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$A_5:D_6^2$

$\rhd$

$A_5:D_6^2$

$\rhd$

$C_3\times \GL(2,4)$

$\rhd$

$C_3\times \GL(2,4)$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 20 larger groups in the database.

This group is a maximal quotient of 5 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 $126 \times 126$ rational character table. Alternatively, you may search for characters of this group with desired properties.