Abstract group 11520.da: $D_4\times A_4\times S_5$

• Label: 11520.da
• Order: \( 2^{8} \cdot 3^{2} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{2} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $13$
• Trans deg.: $60$
• Rank: $3$

Group information

Description:	$D_4\times A_4\times S_5$
Order:	\(11520\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$(S_5\times \GL(2,\mathbb{Z}/4)).C_2^2$, of order \(46080\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 5 \)
Composition factors:	$C_2$ x 6, $C_3$, $A_5$
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	623	188	1168	24	3940	552	3296	192	192	960	384	11520
Conjugacy classes	1	23	5	16	1	51	7	24	2	2	6	2	140
Divisions	1	23	3	16	1	33	7	14	1	2	3	1	105
Autjugacy classes	1	15	3	12	1	20	5	12	1	2	2	1	75

Dimension	1	2	3	4	5	6	8	10	12	15	16	18	20	24	30	36
Irr. complex chars.	24	6	8	24	24	14	6	6	11	8	0	4	0	2	2	1	140
Irr. rational chars.	8	10	8	10	8	6	10	10	13	8	2	4	2	3	2	1	105

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$60$
Rank:	$3$
Inequivalent generating triples:	$9107280$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $13$ $\langle(6,7)(8,9)(10,11)(12,13), (6,7)(8,9)(10,12), (6,8)(7,9)(11,13), (6,7)(8,9) \!\cdots\! \rangle$
Direct product:	$D_4$ $\, \times\, $ $A_4$ $\, \times\, $ $S_5$
Semidirect product:	$(C_2^4\times S_5)$ $\,\rtimes\,$ $C_6$	$(C_2^4:S_5)$ $\,\rtimes\,$ $C_6$	$C_2^4$ $\,\rtimes\,$ $(C_6\times S_5)$	$A_5$ $\,\rtimes\,$ $(C_2^5:C_6)$	all 40
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times A_4\times S_5)$ . $C_2^2$	$C_2$ . $(C_2^2\times A_4\times S_5)$	$(C_2\times S_5)$ . $(C_2^2\times A_4)$	$(C_2\times A_4)$ . $(C_2^2\times S_5)$	all 9

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

Homology

Abelianization:	$C_{2}^{2} \times C_{6} \simeq C_{2}^{3} \times C_{3}$
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 334516 subgroups in 6827 conjugacy classes, 75 normal (39 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^2\times A_4\times S_5$
Commutator:	$G' \simeq$ $C_2^3\times A_5$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^2\times A_4\times S_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times D_4$	$G/\operatorname{Fit} \simeq$ $C_3\times S_5$
Radical:	$R \simeq$ $D_4\times A_4$	$G/R \simeq$ $S_5$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times A_5$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
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 11520: $D_4\times A_4\times S_5$

Order 5760: $A_4\times C_2^2:S_5$ x 2, $C_2^2\times A_4\times S_5$ x 2, $D_4\times A_4\times A_5$, $A_4\times C_4:S_5$, $C_4\times A_4\times S_5$

Order 3840: $C_2^2\times D_4\times S_5$

Order 2880: $C_2\times A_4\times S_5$ x 7, $C_2^4:\GL(2,4)$ x 2, $C_4\times A_4\times A_5$, $C_3\times D_4\times S_5$, $A_4\times A_5:C_4$

Order 2304: $D_4\times A_4\times S_4$

Order 1920: $C_2\times D_4\times S_5$ x 8, $C_2^4\times S_5$ x 2, $C_2^4:S_5$ x 2, $C_2^2\times D_4\times A_5$, $C_2^2\times C_4:S_5$, $C_4\times C_2^2\times S_5$, $D_4\times A_4\times F_5$

Order 1440: $A_4\times S_5$ x 4, $C_2^3:\GL(2,4)$ x 3, $\GL(2,4):C_2^3$ x 2, $\GL(2,4):D_4$ x 2, $D_4\times \GL(2,4)$, $C_{12}:S_5$, $C_{12}\times S_5$

Order 1152: $A_4^2:C_2^3$ x 2, $A_4\times \GL(2,\mathbb{Z}/4)$ x 2, $D_4\times A_4\times D_6$, $D_4\times A_4^2$, $A_4^2:D_4$, $C_4\times A_4\times S_4$

Order 960: $D_4\times S_5$ x 22, $C_2^3\times S_5$ x 19, $C_2^3:S_5$ x 8, $C_2\times D_4\times A_5$ x 4, $C_2\times C_4:S_5$ x 4, $C_2\times C_4\times S_5$ x 4, $C_2^2\times A_4\times F_5$ x 2, $C_2^4\times A_5$ x 2, $A_4\times D_{10}:C_4$ x 2, $D_4\times D_5\times A_4$, $A_4\times C_{20}:C_4$, $C_4\times A_4\times F_5$, $C_4\times C_2^2\times A_5$, $C_2^3.S_5$

Order 768: $\GL(2,\mathbb{Z}/4):C_2^3$, $A_4\times D_4^2$

Order 720: $C_6\times S_5$ x 7, $C_2^2\times \GL(2,4)$ x 2, $A_4\times A_5$, $C_{12}\times A_5$, $A_5:C_{12}$

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

Order 576: $S_3\times D_4\times A_4$ x 8, $A_4^2:C_2^2$ x 7, $C_2^2\times A_4\times D_6$ x 2, $C_2^2\times A_4^2$ x 2, $C_6:D_4\times A_4$ x 2, $C_2^4:C_6^2$, $C_2\times A_4\times D_{12}$, $C_4\times A_4\times D_6$, $\GL(2,\mathbb{Z}/4):C_6$, $C_4\times A_4^2$, $A_4^2:C_4$

Order 480: $C_2^2\times S_5$ x 58, $C_2^2:S_5$ x 12, $C_2^3\times A_5$ x 9, $C_2\times A_4\times F_5$ x 8, $D_4\times A_5$ x 6, $C_4:S_5$ x 6, $C_4\times S_5$ x 6, $C_2\times C_4\times A_5$ x 2, $C_2^2.S_5$ x 2, $C_2\times A_4\times D_{10}$ x 2, $A_4\times C_5:D_4$ x 2, $C_2^4:C_{30}$, $D_{20}:C_{12}$, $A_4\times D_{20}$, $C_4\times D_5\times A_4$

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$ x 8, $C_2^6:C_6$ x 5, $A_4\times C_4:D_4$ x 4, $C_2^6:C_6$ x 4, $C_2^4\times S_4$ x 2, $C_2^2\times \GL(2,\mathbb{Z}/4)$ x 2, $C_4\times D_4\times A_4$ x 2, $C_{12}:C_2^5$, $C_2^6:C_6$, $C_2^4:D_{12}$, $C_2^5.D_6$, $A_4\times C_4:D_4$

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

Order 320: $D_{10}.C_2^4$ x 8, $C_2^4\times F_5$ x 2, $C_2^4:F_5$ x 2, $C_{20}:C_2^4$, $D_{10}.C_2^4$, $D_{10}:C_4^2$

Order 288: $C_2\times A_4\times D_6$ x 19, $A_4\times C_3:D_4$ x 8, $C_2^3.C_6^2$ x 5, $A_4\times D_{12}$ x 4, $C_4\times S_3\times A_4$ x 4, $A_4\times S_4$ x 4, $C_2\times A_4^2$ x 3, $C_3\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^3:C_6^2$ x 2, $C_2\times C_6\times S_4$ x 2, $C_6^2:C_2^3$, $C_2^3.C_6^2$, $C_6:C_4\times A_4$, $C_{12}:S_4$, $C_{12}\times S_4$

Order 256: $C_2^2\times D_4^2$

Order 240: $C_2\times S_5$ x 46, $C_2^2\times A_5$ x 15, $A_4\times D_{10}$ x 7, $A_4\times F_5$ x 5, $C_4\times A_5$ x 2, $A_5:C_4$ x 2, $C_2^3:C_{30}$ x 2, $D_{10}:C_{12}$ x 2, $D_{10}:C_{12}$ x 2, $C_{12}:D_{10}$, $A_4\times C_{20}$, $C_{20}:C_{12}$, $C_{12}\times F_5$, $A_4\times C_5:C_4$

Order 192: $C_2^5:C_6$ x 33, $C_{12}:C_2^4$ x 24, $D_4\times S_4$ x 22, $C_2^3\times S_4$ x 19, $C_2^4:C_{12}$ x 8, $C_2\times \GL(2,\mathbb{Z}/4)$ x 8, $A_4\times C_2^4$ x 6, $C_2^4:C_{12}$ x 5, $C_2^3:D_{12}$ x 4, $C_2^4.D_6$ x 4, $A_4\times C_4:C_4$ x 2, $D_6\times C_2^4$ x 2, $C_2^2\wr C_3$ x 2, $D_6:C_2^4$ x 2, $A_4\times C_4^2$, $C_{12}:C_2^4$, $C_2^3\times D_{12}$, $C_{12}:C_2^4$, $C_2^4:C_{12}$, $C_2^3.S_4$, $C_3\times D_4^2$

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

Order 160: $D_4\times F_5$ x 22, $C_2^3\times F_5$ x 20, $D_4\times D_{10}$ x 8, $C_2^3:F_5$ x 8, $D_{10}.D_4$ x 4, $C_{10}:C_4^2$ x 4, $C_2^3\times D_{10}$ 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 144: $A_4\times D_6$ x 30, $C_2^2:C_6^2$ x 11, $C_{12}:D_6$ x 8, $C_6\times S_4$ x 7, $C_{12}\times A_4$ x 3, $C_6^2:C_2^2$ x 2, $C_6^2:C_2^2$ x 2, $A_4\times C_3:C_4$ x 2, $A_4^2$, $C_4:C_6^2$, $C_6\times D_{12}$, $C_{12}\times D_6$, $A_4:C_{12}$

Order 128: $C_2\times D_4^2$ x 16, $D_4\times C_2^4$ x 4, $C_2^4:D_4$ x 4, $C_2^4:D_4$ x 4, $C_4^2:C_2^3$ x 2, $C_4^2:C_2^3$

Order 120: $C_6\times F_5$ x 8, $S_5$ x 8, $C_2\times A_5$ x 7, $D_5\times A_4$ x 4, $C_{10}\times A_4$ x 3, $C_6\times D_{10}$ x 2, $C_{15}:D_4$ x 2, $D_4\times C_{15}$, $C_3\times D_{20}$, $D_5\times C_{12}$

Order 96: $D_4\times D_6$ x 150, $C_2^2\times S_4$ x 58, $C_2^3\times D_6$ x 43, $D_4\times A_4$ x 41, $C_2^3\times A_4$ x 33, $C_2^3:D_6$ x 24, $C_{12}:C_2^3$ x 16, $C_2^3:C_{12}$ x 16, $C_2^2\times D_{12}$ x 12, $C_{12}:C_2^3$ x 12, $\GL(2,\mathbb{Z}/4)$ x 12, $C_4:S_4$ x 6, $C_4\times S_4$ x 6, $C_{12}:D_4$ x 4, $C_2^4:C_6$ x 4, $C_2^3:A_4$ x 3, $C_2^4\times C_6$ x 2, $C_2^2.S_4$ x 2, $D_4\times C_{12}$ x 2, $C_2^3\times C_{12}$, $C_6.C_2^4$, $C_{12}:D_4$

Order 80: $C_2^2\times F_5$ x 62, $D_4\times D_5$ x 22, $C_2^2\times D_{10}$ x 19, $D_{10}:C_4$ x 12, $C_{10}:D_4$ x 8, $C_{20}:C_4$ x 6, $C_4\times F_5$ x 6, $D_4\times C_{10}$ 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 72: $C_6\times D_6$ x 19, $C_6\times A_4$ x 10, $S_3\times A_4$ x 8, $C_6\wr C_2$ x 8, $C_3\times S_4$ x 4, $D_4\times C_3^2$ x 4, $C_3\times D_{12}$ x 4, $S_3\times C_{12}$ x 4, $C_2\times C_6^2$ x 2, $C_6\times C_{12}$, $C_6:C_{12}$

Order 64: $D_4^2$ x 86, $D_4\times C_2^3$ x 72, $C_2^3:D_4$ x 32, $C_2^3:D_4$ x 32, $C_4^2:C_2^2$ x 16, $C_4^2:C_2^2$ x 8, $C_2^4:C_4$ x 8, $C_2^6$ x 4, $C_2^4\times C_4$ x 4, $C_2^3.D_4$ x 2, $C_2^2\times C_4^2$

Order 60: $C_3\times D_{10}$ x 7, $C_3\times F_5$ x 5, $C_2\times C_{30}$ x 2, $C_5\times A_4$, $A_5$, $C_{60}$, $C_5:C_{12}$

Order 48: $C_2^2\times D_6$ x 270, $S_3\times D_4$ x 176, $C_6:D_4$ x 76, $C_6\times D_4$ x 67, $C_2^2\times A_4$ x 61, $C_2\times S_4$ x 46, $C_2\times D_{12}$ x 38, $C_4\times D_6$ x 38, $C_2^3\times C_6$ x 25, $C_4\times A_4$ x 14, $C_2^2\times C_{12}$ x 10, $C_2^2:C_{12}$ x 8, $C_6.C_2^3$ x 6, $A_4:C_4$ x 2, $C_4:C_{12}$ x 2, $C_2^2:A_4$, $C_4\times C_{12}$

Order 40: $C_2\times D_{10}$ x 58, $C_2\times F_5$ x 52, $C_5:D_4$ x 12, $C_2^2\times C_{10}$ x 9, $D_{20}$ 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: $C_6\times S_3$ x 30, $C_6^2$ x 9, $C_3\times C_{12}$ x 2, $C_3:C_{12}$ x 2, $C_3\times A_4$ x 2

Order 32: $C_2^2\times D_4$ x 413, $C_4:D_4$ x 88, $C_2^2\wr C_2$ x 88, $C_2^5$ x 52, $C_4\times D_4$ x 44, $C_2^3\times C_4$ x 35, $C_2^3:C_4$ x 32, $C_4:D_4$ x 22, $C_2^2.D_4$ x 8, $C_2\times C_4^2$ x 4

Order 30: $C_3\times D_5$ x 4, $C_{30}$ x 3

Order 24: $C_2\times D_6$ x 462, $C_2^2\times C_6$ x 91, $C_3\times D_4$ x 57, $C_3:D_4$ x 48, $C_2\times A_4$ x 31, $C_2\times C_{12}$ x 24, $D_{12}$ x 24, $C_4\times S_3$ x 24, $C_6:C_4$ x 10, $S_4$ x 8

Order 20: $D_{10}$ x 46, $C_2\times C_{10}$ x 15, $F_5$ x 10, $C_{20}$ x 2, $C_5:C_4$ x 2

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

Order 16: $C_2\times D_4$ x 612, $C_2^4$ x 263, $C_2^2\times C_4$ x 105, $C_2^2:C_4$ x 48, $C_4:C_4$ x 12, $C_4^2$ x 6

Order 15: $C_{15}$

Order 12: $D_6$ x 188, $C_2\times C_6$ x 103, $C_{12}$ x 14, $A_4$ x 5, $C_3:C_4$ x 4

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

Order 9: $C_3^2$

Order 8: $C_2^3$ x 414, $D_4$ x 174, $C_2\times C_4$ x 84

Order 6: $C_6$ x 33, $S_3$ x 16

Order 5: $C_5$

Order 4: $C_2^2$ x 184, $C_4$ x 16

Order 3: $C_3$ x 3

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 11520: $D_4\times A_4\times S_5$

Order 5760: $D_4\times A_4\times A_5$, $A_4\times C_4:S_5$, $C_4\times A_4\times S_5$, $A_4\times C_2^2:S_5$, $C_2^2\times A_4\times S_5$

Order 3840: $C_2^2\times D_4\times S_5$

Order 2880: $C_2\times A_4\times S_5$ x 3, $C_4\times A_4\times A_5$, $C_2^4:\GL(2,4)$, $C_3\times D_4\times S_5$, $A_4\times A_5:C_4$

Order 2304: $D_4\times A_4\times S_4$

Order 1920: $C_2\times D_4\times S_5$ x 6, $C_2^4\times S_5$, $C_2^2\times D_4\times A_5$, $C_2^2\times C_4:S_5$, $C_2^4:S_5$, $C_4\times C_2^2\times S_5$, $D_4\times A_4\times F_5$

Order 1440: $C_2^3:\GL(2,4)$ x 2, $A_4\times S_5$ x 2, $\GL(2,4):C_2^3$, $D_4\times \GL(2,4)$, $\GL(2,4):D_4$, $C_{12}:S_5$, $C_{12}\times S_5$

Order 1152: $A_4^2:C_2^3$, $D_4\times A_4\times D_6$, $D_4\times A_4^2$, $A_4\times \GL(2,\mathbb{Z}/4)$, $A_4^2:D_4$, $C_4\times A_4\times S_4$

Order 960: $D_4\times S_5$ x 11, $C_2^3\times S_5$ x 8, $C_2^3:S_5$ x 4, $C_2\times C_4\times S_5$ x 4, $C_2\times D_4\times A_5$ x 3, $C_2\times C_4:S_5$ x 3, $C_2^2\times A_4\times F_5$, $C_2^4\times A_5$, $D_4\times D_5\times A_4$, $A_4\times D_{10}:C_4$, $A_4\times C_{20}:C_4$, $C_4\times A_4\times F_5$, $C_4\times C_2^2\times A_5$, $C_2^3.S_5$

Order 768: $\GL(2,\mathbb{Z}/4):C_2^3$, $A_4\times D_4^2$

Order 720: $C_6\times S_5$ x 3, $C_2^2\times \GL(2,4)$, $A_4\times A_5$, $C_{12}\times A_5$, $A_5:C_{12}$

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

Order 576: $S_3\times D_4\times A_4$ x 6, $A_4^2:C_2^2$ x 3, $C_2^2\times A_4\times D_6$, $C_2^2\times A_4^2$, $C_2^4:C_6^2$, $C_6:D_4\times A_4$, $C_2\times A_4\times D_{12}$, $C_4\times A_4\times D_6$, $\GL(2,\mathbb{Z}/4):C_6$, $C_4\times A_4^2$, $A_4^2:C_4$

Order 480: $C_2^2\times S_5$ x 21, $C_2^2:S_5$ x 5, $C_4\times S_5$ x 5, $C_2^3\times A_5$ x 5, $D_4\times A_5$ x 4, $C_4:S_5$ x 4, $C_2\times A_4\times F_5$ x 4, $C_2\times C_4\times A_5$ x 2, $C_2^2.S_5$ x 2, $C_2\times A_4\times D_{10}$, $C_2^4:C_{30}$, $D_{20}:C_{12}$, $A_4\times C_5:D_4$, $A_4\times D_{20}$, $C_4\times D_5\times A_4$

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$ x 6, $C_2^6:C_6$ x 4, $A_4\times C_4:D_4$ x 3, $C_2^6:C_6$ x 2, $C_4\times D_4\times A_4$ x 2, $C_2^4\times S_4$, $C_{12}:C_2^5$, $C_2^6:C_6$, $C_2^2\times \GL(2,\mathbb{Z}/4)$, $C_2^4:D_{12}$, $C_2^5.D_6$, $A_4\times C_4:D_4$

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

Order 320: $D_{10}.C_2^4$ x 6, $C_2^4\times F_5$, $C_{20}:C_2^4$, $C_2^4:F_5$, $D_{10}.C_2^4$, $D_{10}:C_4^2$

Order 288: $C_2\times A_4\times D_6$ x 8, $C_2^3.C_6^2$ x 4, $A_4\times C_3:D_4$ x 4, $C_4\times S_3\times A_4$ x 4, $A_4\times D_{12}$ x 3, $C_2\times A_4^2$ x 2, $A_4\times S_4$ x 2, $C_6^2:C_2^3$, $C_2^3.C_6^2$, $C_6:C_4\times A_4$, $C_3\times \GL(2,\mathbb{Z}/4)$, $C_{12}:S_4$, $C_{12}\times S_4$, $C_2^3:C_6^2$, $C_2\times C_6\times S_4$

Order 256: $C_2^2\times D_4^2$

Order 240: $C_2\times S_5$ x 17, $C_2^2\times A_5$ x 9, $A_4\times D_{10}$ x 4, $A_4\times F_5$ x 3, $C_4\times A_5$ x 2, $A_5:C_4$ x 2, $C_2^3:C_{30}$, $D_{10}:C_{12}$, $C_{12}:D_{10}$, $A_4\times C_{20}$, $D_{10}:C_{12}$, $C_{20}:C_{12}$, $C_{12}\times F_5$, $A_4\times C_5:C_4$

Order 192: $C_2^5:C_6$ x 19, $C_{12}:C_2^4$ x 18, $D_4\times S_4$ x 11, $C_2^3\times S_4$ x 8, $C_2^4:C_{12}$ x 5, $C_2^4:C_{12}$ x 4, $C_2\times \GL(2,\mathbb{Z}/4)$ x 4, $C_2^4.D_6$ x 4, $A_4\times C_2^4$ x 3, $C_2^3:D_{12}$ x 3, $A_4\times C_4:C_4$ x 2, $A_4\times C_4^2$, $D_6\times C_2^4$, $C_2^2\wr C_3$, $C_{12}:C_2^4$, $D_6:C_2^4$, $C_2^3\times D_{12}$, $C_{12}:C_2^4$, $C_2^4:C_{12}$, $C_2^3.S_4$, $C_3\times D_4^2$

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

Order 160: $D_4\times F_5$ x 11, $C_2^3\times F_5$ x 9, $D_4\times D_{10}$ x 6, $C_2^3:F_5$ x 4, $C_{10}:C_4^2$ x 4, $D_{10}.D_4$ x 3, $C_2^3\times D_{10}$, $C_{20}:C_2^3$, $C_2^3:D_{10}$, $C_2^2\times D_{20}$, $C_{20}:C_2^3$

Order 144: $A_4\times D_6$ x 13, $C_{12}:D_6$ x 6, $C_2^2:C_6^2$ x 5, $C_6\times S_4$ x 3, $C_{12}\times A_4$ x 3, $A_4\times C_3:C_4$ x 2, $C_6^2:C_2^2$, $A_4^2$, $C_4:C_6^2$, $C_6^2:C_2^2$, $C_6\times D_{12}$, $C_{12}\times D_6$, $A_4:C_{12}$

Order 128: $C_2\times D_4^2$ x 12, $D_4\times C_2^4$ x 3, $C_2^4:D_4$ x 3, $C_2^4:D_4$ x 2, $C_4^2:C_2^3$ x 2, $C_4^2:C_2^3$

Order 120: $C_2\times A_5$ x 5, $C_6\times F_5$ x 4, $S_5$ x 4, $D_5\times A_4$ x 3, $C_{10}\times A_4$ x 2, $C_6\times D_{10}$, $D_4\times C_{15}$, $C_{15}:D_4$, $C_3\times D_{20}$, $D_5\times C_{12}$

Order 96: $D_4\times D_6$ x 75, $D_4\times A_4$ x 24, $C_2^2\times S_4$ x 21, $C_2^3\times D_6$ x 18, $C_2^3\times A_4$ x 17, $C_{12}:C_2^3$ x 12, $C_2^3:D_6$ x 12, $C_{12}:C_2^3$ x 12, $C_2^3:C_{12}$ x 12, $C_2^2\times D_{12}$ x 9, $\GL(2,\mathbb{Z}/4)$ x 5, $C_4\times S_4$ x 5, $C_4:S_4$ x 4, $C_{12}:D_4$ x 3, $C_2^3:A_4$ x 2, $C_2^2.S_4$ x 2, $C_2^4:C_6$ x 2, $D_4\times C_{12}$ x 2, $C_2^4\times C_6$, $C_2^3\times C_{12}$, $C_6.C_2^4$, $C_{12}:D_4$

Order 80: $C_2^2\times F_5$ x 25, $D_4\times D_5$ x 11, $C_2^2\times D_{10}$ x 10, $D_{10}:C_4$ x 5, $C_4\times F_5$ x 5, $C_{10}:D_4$ x 4, $C_4\times D_{10}$ x 4, $C_{20}:C_4$ x 4, $D_4\times C_{10}$ x 3, $C_2\times D_{20}$ x 3, $C_2^3\times C_{10}$, $C_2^2\times C_{20}$, $C_2^2.D_{10}$

Order 72: $C_6\times D_6$ x 8, $C_6\times A_4$ x 6, $S_3\times A_4$ x 5, $C_6\wr C_2$ x 4, $S_3\times C_{12}$ x 4, $D_4\times C_3^2$ x 3, $C_3\times D_{12}$ x 3, $C_3\times S_4$ x 2, $C_2\times C_6^2$, $C_6\times C_{12}$, $C_6:C_{12}$

Order 64: $D_4\times C_2^3$ x 40, $D_4^2$ x 35, $C_2^3:D_4$ x 20, $C_2^3:D_4$ x 16, $C_4^2:C_2^2$ x 14, $C_4^2:C_2^2$ x 6, $C_2^4:C_4$ x 5, $C_2^4\times C_4$ x 3, $C_2^6$ x 2, $C_2^3.D_4$ x 2, $C_2^2\times C_4^2$

Order 60: $C_3\times D_{10}$ x 4, $C_3\times F_5$ x 3, $C_5\times A_4$, $A_5$, $C_{60}$, $C_5:C_{12}$, $C_2\times C_{30}$

Order 48: $C_2^2\times D_6$ x 89, $S_3\times D_4$ x 74, $C_6\times D_4$ x 37, $C_2^2\times A_4$ x 32, $C_6:D_4$ x 31, $C_4\times D_6$ x 31, $C_2\times D_{12}$ x 21, $C_2\times S_4$ x 17, $C_4\times A_4$ x 12, $C_2^3\times C_6$ x 11, $C_2^2\times C_{12}$ x 9, $C_6.C_2^3$ x 6, $C_2^2:C_{12}$ x 5, $A_4:C_4$ x 2, $C_4:C_{12}$ x 2, $C_2^2:A_4$, $C_4\times C_{12}$

Order 40: $C_2\times D_{10}$ x 29, $C_2\times F_5$ x 23, $C_5:D_4$ x 5, $C_4\times D_5$ x 5, $C_2^2\times C_{10}$ x 5, $D_{20}$ x 4, $C_5\times D_4$ x 4, $C_2\times C_{20}$ x 2, $C_{10}:C_4$ x 2

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

Order 32: $C_2^2\times D_4$ x 173, $C_4:D_4$ x 37, $C_2^2\wr C_2$ x 30, $C_4\times D_4$ x 26, $C_2^5$ x 25, $C_2^3\times C_4$ x 24, $C_2^3:C_4$ x 20, $C_4:D_4$ x 11, $C_2^2.D_4$ x 7, $C_2\times C_4^2$ x 4

Order 30: $C_3\times D_5$ x 3, $C_{30}$ x 2

Order 24: $C_2\times D_6$ x 144, $C_2^2\times C_6$ x 38, $C_3\times D_4$ x 31, $C_3:D_4$ x 20, $C_4\times S_3$ x 20, $C_2\times A_4$ x 20, $C_2\times C_{12}$ x 19, $D_{12}$ x 13, $C_6:C_4$ x 9, $S_4$ x 4

Order 20: $D_{10}$ x 25, $C_2\times C_{10}$ x 9, $F_5$ x 6, $C_{20}$ x 2, $C_5:C_4$ x 2

Order 18: $C_3\times S_3$ x 5, $C_3\times C_6$ x 4

Order 16: $C_2\times D_4$ x 222, $C_2^4$ x 110, $C_2^2\times C_4$ x 64, $C_2^2:C_4$ x 25, $C_4:C_4$ x 9, $C_4^2$ x 5

Order 15: $C_{15}$

Order 12: $D_6$ x 72, $C_2\times C_6$ x 47, $C_{12}$ x 12, $A_4$ x 5, $C_3:C_4$ x 4

Order 10: $D_5$ x 6, $C_{10}$ x 5

Order 9: $C_3^2$

Order 8: $C_2^3$ x 165, $D_4$ x 70, $C_2\times C_4$ x 52

Order 6: $C_6$ x 20, $S_3$ x 10

Order 5: $C_5$

Order 4: $C_2^2$ x 87, $C_4$ x 12

Order 3: $C_3$ x 3

Order 2: $C_2$ x 15

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 11520: $D_4\times A_4\times S_5$ ($C_1$)

Order 5760: $A_4\times C_2^2:S_5$ ($C_2$) x 2, $C_2^2\times A_4\times S_5$ ($C_2$) x 2, $D_4\times A_4\times A_5$ ($C_2$), $A_4\times C_4:S_5$ ($C_2$), $C_4\times A_4\times S_5$ ($C_2$)

Order 3840: $C_2^2\times D_4\times S_5$ ($C_3$)

Order 2880: $C_2\times A_4\times S_5$ ($C_2^2$) x 3, $C_2^4:\GL(2,4)$ ($C_2^2$) x 2, $C_4\times A_4\times A_5$ ($C_2^2$), $A_4\times A_5:C_4$ ($C_2^2$)

Order 1920: $C_2^4\times S_5$ ($C_6$) x 2, $C_2^4:S_5$ ($C_6$) x 2, $C_2^2\times D_4\times A_5$ ($C_6$), $C_2^2\times C_4:S_5$ ($C_6$), $C_4\times C_2^2\times S_5$ ($C_6$)

Order 1440: $A_4\times S_5$ ($D_4$) x 2, $C_2^3:\GL(2,4)$ ($C_2^3$)

Order 960: $C_2^3\times S_5$ ($C_2\times C_6$) x 3, $C_2^4\times A_5$ ($C_2\times C_6$) x 2, $C_4\times C_2^2\times A_5$ ($C_2\times C_6$), $C_2^3.S_5$ ($C_2\times C_6$), $D_4\times S_5$ ($A_4$)

Order 720: $A_4\times A_5$ ($C_2\times D_4$)

Order 480: $C_2^2:S_5$ ($C_2\times A_4$) x 2, $C_2^2\times S_5$ ($C_2\times A_4$) x 2, $C_2^2\times S_5$ ($C_3\times D_4$) x 2, $D_4\times A_5$ ($C_2\times A_4$), $C_4:S_5$ ($C_2\times A_4$), $C_4\times S_5$ ($C_2\times A_4$), $C_2^3\times A_5$ ($C_2^2\times C_6$)

Order 240: $C_2\times S_5$ ($C_2^2\times A_4$) x 3, $C_2^2\times A_5$ ($C_2^2\times A_4$) x 2, $C_4\times A_5$ ($C_2^2\times A_4$), $A_5:C_4$ ($C_2^2\times A_4$), $C_2^2\times A_5$ ($C_6\times D_4$)

Order 120: $S_5$ ($D_4\times A_4$) x 2, $C_2\times A_5$ ($C_2^3\times A_4$)

Order 96: $D_4\times A_4$ ($S_5$)

Order 60: $A_5$ ($C_2^5:C_6$)

Order 48: $C_2^2\times A_4$ ($C_2\times S_5$) x 2, $C_4\times A_4$ ($C_2\times S_5$)

Order 32: $C_2^2\times D_4$ ($C_3\times S_5$)

Order 24: $C_2\times A_4$ ($C_2^2\times S_5$)

Order 16: $C_2^4$ ($C_6\times S_5$) x 2, $C_2^2\times C_4$ ($C_6\times S_5$)

Order 12: $A_4$ ($D_4\times S_5$)

Order 8: $C_2^3$ ($\GL(2,4):C_2^3$), $D_4$ ($A_4\times S_5$)

Order 4: $C_2^2$ ($C_2\times A_4\times S_5$) x 2, $C_2^2$ ($C_3\times D_4\times S_5$), $C_4$ ($C_2\times A_4\times S_5$)

Order 2: $C_2$ ($C_2^2\times A_4\times S_5$)

Order 1: $C_1$ ($D_4\times A_4\times S_5$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 11520: $D_4\times A_4\times S_5$ ($C_1$)

Order 5760: $D_4\times A_4\times A_5$ ($C_2$), $A_4\times C_4:S_5$ ($C_2$), $C_4\times A_4\times S_5$ ($C_2$), $A_4\times C_2^2:S_5$ ($C_2$), $C_2^2\times A_4\times S_5$ ($C_2$)

Order 3840: $C_2^2\times D_4\times S_5$ ($C_3$)

Order 2880: $C_2\times A_4\times S_5$ ($C_2^2$) x 2, $C_4\times A_4\times A_5$ ($C_2^2$), $C_2^4:\GL(2,4)$ ($C_2^2$), $A_4\times A_5:C_4$ ($C_2^2$)

Order 1920: $C_2^4\times S_5$ ($C_6$), $C_2^2\times D_4\times A_5$ ($C_6$), $C_2^2\times C_4:S_5$ ($C_6$), $C_2^4:S_5$ ($C_6$), $C_4\times C_2^2\times S_5$ ($C_6$)

Order 1440: $C_2^3:\GL(2,4)$ ($C_2^3$), $A_4\times S_5$ ($D_4$)

Order 960: $C_2^3\times S_5$ ($C_2\times C_6$) x 2, $C_2^4\times A_5$ ($C_2\times C_6$), $C_4\times C_2^2\times A_5$ ($C_2\times C_6$), $C_2^3.S_5$ ($C_2\times C_6$), $D_4\times S_5$ ($A_4$)

Order 720: $A_4\times A_5$ ($C_2\times D_4$)

Order 480: $D_4\times A_5$ ($C_2\times A_4$), $C_2^2:S_5$ ($C_2\times A_4$), $C_4:S_5$ ($C_2\times A_4$), $C_4\times S_5$ ($C_2\times A_4$), $C_2^3\times A_5$ ($C_2^2\times C_6$), $C_2^2\times S_5$ ($C_2\times A_4$), $C_2^2\times S_5$ ($C_3\times D_4$)

Order 240: $C_2\times S_5$ ($C_2^2\times A_4$) x 2, $C_4\times A_5$ ($C_2^2\times A_4$), $A_5:C_4$ ($C_2^2\times A_4$), $C_2^2\times A_5$ ($C_2^2\times A_4$), $C_2^2\times A_5$ ($C_6\times D_4$)

Order 120: $C_2\times A_5$ ($C_2^3\times A_4$), $S_5$ ($D_4\times A_4$)

Order 96: $D_4\times A_4$ ($S_5$)

Order 60: $A_5$ ($C_2^5:C_6$)

Order 48: $C_2^2\times A_4$ ($C_2\times S_5$), $C_4\times A_4$ ($C_2\times S_5$)

Order 32: $C_2^2\times D_4$ ($C_3\times S_5$)

Order 24: $C_2\times A_4$ ($C_2^2\times S_5$)

Order 16: $C_2^4$ ($C_6\times S_5$), $C_2^2\times C_4$ ($C_6\times S_5$)

Order 12: $A_4$ ($D_4\times S_5$)

Order 8: $C_2^3$ ($\GL(2,4):C_2^3$), $D_4$ ($A_4\times S_5$)

Order 4: $C_2^2$ ($C_3\times D_4\times S_5$), $C_2^2$ ($C_2\times A_4\times S_5$), $C_4$ ($C_2\times A_4\times S_5$)

Order 2: $C_2$ ($C_2^2\times A_4\times S_5$)

Order 1: $C_1$ ($D_4\times A_4\times S_5$)

Series

Derived series

$D_4\times A_4\times S_5$

$\rhd$

$D_4\times A_4\times S_5$

$\rhd$

$C_2^3\times A_5$

$\rhd$

$C_2^3\times A_5$

$\rhd$

$A_5$

$\rhd$

$A_5$

Chief series

$D_4\times A_4\times S_5$

$\rhd$

$D_4\times A_4\times S_5$

$\rhd$

$D_4\times A_4\times A_5$

$\rhd$

$D_4\times A_4\times A_5$

$\rhd$

$D_4\times A_4$

$\rhd$

$D_4\times A_4$

$\rhd$

$C_2^2\times A_4$

$\rhd$

$C_2^2\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$D_4\times A_4\times S_5$

$\rhd$

$D_4\times A_4\times S_5$

$\rhd$

$C_2^3\times A_5$

$\rhd$

$C_2^3\times A_5$

$\rhd$

$C_2^2\times A_5$

$\rhd$

$C_2^2\times A_5$

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 8 larger groups in the database.

This group is a maximal quotient of 2 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.