Abstract group 23040.g: $C_2^2\times A_4\times \GL(2,5)$

• Label: 23040.g
• Order: \( 2^{9} \cdot 3^{2} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{3} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \cdot 3 \)
• Perm deg.: $32$
• Trans deg.: $288$
• Rank: $3$

Group information

Description:	$C_2^2\times A_4\times \GL(2,5)$
Order:	\(23040\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 5 \)
Exponent:	\(120\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \)
Automorphism group:	$C_2^3.A_4^2.C_2^3.S_5$, of order \(1105920\)\(\medspace = 2^{13} \cdot 3^{3} \cdot 5 \)
Composition factors:	$C_2$ x 7, $C_3$, $A_5$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	10	12	15	20	24	30	60
Elements	1	511	188	2432	24	2756	640	744	6784	192	768	5120	1344	1536	23040
Conjugacy classes	1	23	5	56	1	51	16	15	88	2	16	80	14	16	384
Divisions	1	23	3	32	1	33	8	15	28	1	8	16	7	4	180
Autjugacy classes	1	9	3	10	1	11	2	5	11	1	4	4	2	2	66

Dimension	1	2	3	4	5	6	8	10	12	15	16	18	20	24	30	32	36	48
Irr. complex chars.	48	0	16	120	48	72	0	0	40	16	0	24	0	0	0	0	0	0	384
Irr. rational chars.	8	12	8	12	8	12	16	12	24	8	12	8	4	16	4	4	8	4	180

Minimal presentations

Permutation degree:	$32$
Transitive degree:	$288$
Rank:	$3$
Inequivalent generating triples:	$3035760$

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 $32$ $\langle(2,9)(4,16)(5,6)(7,21)(8,14)(10,20)(11,19)(13,22)(15,23)(17,24), (25,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 7 & 12 \\ 4 & 3 \end{array}\right), \left(\begin{array}{rr} 8 & 11 \\ 17 & 11 \end{array}\right), \left(\begin{array}{rr} 7 & 6 \\ 2 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $A_4$ $\, \times\, $ $\GL(2,5)$
Semidirect product:	$(C_2^4\times \GL(2,5))$ $\,\rtimes\,$ $C_3$	$(C_2^5.\GL(2,4))$ $\,\rtimes\,$ $C_4$	$(C_2^3\times \GL(2,5))$ $\,\rtimes\,$ $C_6$	$C_2^4$ $\,\rtimes\,$ $(C_3\times \GL(2,5))$	all 21
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^4:C_{12})$ . $S_5$	$C_2^5$ . $(A_5:C_{12})$	$C_4$ . $(A_4\times C_2^2.S_5)$	$C_4$ . $(C_2^2\times A_4\times S_5)$	all 33

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{20}\Z)$.

Homology

Abelianization:	$C_{2}^{2} \times C_{12} \simeq C_{2}^{2} \times C_{4} \times C_{3}$
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 539802 subgroups in 14590 conjugacy classes, 162 normal (30 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_4$	$G/Z \simeq$ $A_4\times S_5$
Commutator:	$G' \simeq$ $C_2^2\times \SL(2,5)$	$G/G' \simeq$ $C_2^2\times C_{12}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2^2\times A_4\times S_5$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4\times C_4$	$G/\operatorname{Fit} \simeq$ $C_3\times S_5$
Radical:	$R \simeq$ $C_2^4:C_{12}$	$G/R \simeq$ $S_5$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $A_5:C_{12}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:C_2^5$
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 23040: $C_2^2\times A_4\times \GL(2,5)$

Order 11520: $C_2\times A_4\times \GL(2,5)$ x 6, $A_4\times \SL(2,5):C_2^3$

Order 7680: $C_2^4\times \GL(2,5)$

Order 5760: $A_4\times \SL(2,5):C_2^2$ x 6, $A_4\times \GL(2,5)$ x 4, $C_2\times C_6\times \GL(2,5)$, $C_2^5.\GL(2,4)$

Order 4608: $(C_2^4\times \SL(2,3)):C_{12}$

Order 3840: $C_2^3\times \GL(2,5)$ x 14, $\SL(2,5):C_2^5$, $A_4\times D_{10}:C_4^2$

Order 2880: $C_6\times \GL(2,5)$ x 6, $A_4\times \SL(2,5):C_2$ x 4, $C_2^4.\GL(2,4)$ x 3, $C_2^2\times \SL(2,5):C_6$

Order 2304: $(C_2^3\times \SL(2,3)):C_{12}$ x 6, $A_4\times C_{24}:C_2^3$, $A_4\times \SL(2,3):C_2^3$

Order 1920: $C_2^2\times \GL(2,5)$ x 50, $\SL(2,5):C_2^4$ x 14, $A_4\times C_{10}:C_4^2$ x 12, $C_2^3\times A_4\times F_5$ x 2, $C_2^4\times \SL(2,5)$, $C_2^3:C_{12}\times D_{10}$

Order 1536: $C_2^4\times \Unitary(2,3)$, $A_4\times C_4^2:C_2^3$

Order 1440: $C_2\times \SL(2,5):C_6$ x 6, $C_3\times \GL(2,5)$ x 4, $C_2^3.\GL(2,4)$, $A_4\times \SL(2,5)$

Order 1280: $D_{10}.C_2^6$

Order 1152: $C_8:D_6\times A_4$ x 12, $A_4\times \SL(2,3):C_2^2$ x 6, $A_4\times \Unitary(2,3)$ x 4, $C_2^3:C_{12}\times D_6$, $C_2^3.A_4^2$, $(C_2^3\times C_6):C_{24}$, $C_2^2\times \SL(2,3):C_{12}$, $C_2^2\times A_4\times C_{24}$

Order 960: $\SL(2,5):C_2^3$ x 50, $C_2\times \GL(2,5)$ x 44, $C_2^2\times A_4\times F_5$ x 28, $C_4\times A_4\times F_5$ x 16, $C_4\times A_4\times D_{10}$ x 12, $C_2^3\times \SL(2,5)$ x 7, $C_2^2\times A_4\times D_{10}$, $C_2^4:C_{60}$, $C_2^2\times C_{12}\times F_5$, $(C_2^3\times C_{10}):C_{12}$

Order 768: $C_2^3\times \Unitary(2,3)$ x 14, $A_4\times C_4^2:C_2^2$ x 12, $Q_8.(C_2^3\times A_4)$, $\SL(2,3):C_2^5$, $D_4:C_2^3\times A_4$, $C_2^6.C_{12}$, $C_2^2\times A_4\times C_4^2$, $C_{24}:C_2^5$

Order 720: $\SL(2,5):C_6$ x 4, $C_6\times \SL(2,5)$ x 3

Order 640: $D_{10}.C_2^5$ x 28, $F_5\times C_2^5$ x 2, $C_{20}:C_2^5$

Order 576: $A_4\times C_{24}:C_2$ x 16, $C_4\times A_4\times D_6$ x 12, $C_2\times A_4\times C_{24}$ x 6, $C_6:C_8\times A_4$ x 6, $C_6\times \Unitary(2,3)$ x 6, $C_4.A_4^2$ x 4, $C_2^2.A_4^2$ x 3, $C_2^2\times A_4\times D_6$, $C_2^2\times \SL(2,3):C_6$, $C_2^4.C_6^2$, $C_6.C_2^3\times A_4$, $C_6^2:\OD_{16}$

Order 512: $C_4^2:C_2^5$

Order 480: $C_2\times A_4\times F_5$ x 56, $\SL(2,5):C_2^2$ x 44, $C_4\times D_5\times A_4$ x 16, $C_2\times A_4\times D_{10}$ x 14, $C_2^2\times \SL(2,5)$ x 13, $C_{30}:C_4^2$ x 12, $\GL(2,5)$ x 8, $C_2^3:C_{60}$ x 6, $C_{10}:C_4\times A_4$ x 6, $C_2^2\times C_6\times F_5$ x 2, $C_2^4:C_{30}$, $C_{60}:C_2^3$

Order 384: $C_2^2\times \Unitary(2,3)$ x 50, $C_{24}:C_2^4$ x 28, $A_4\times C_4\wr C_2$ x 16, $\SL(2,3):C_2^4$ x 14, $D_4:C_2^2\times A_4$ x 12, $C_2^5.C_{12}$ x 9, $C_2\times A_4\times C_4^2$ x 9, $C_2^4.(C_2\times A_4)$ x 6, $C_2^5:C_{12}$ x 2, $C_2^4:C_{24}$ x 2, $C_{12}:C_2^5$, $C_2^4:\SL(2,3)$, $C_2^2\times Q_8\times A_4$, $C_2^4\times \SL(2,3)$, $C_2^6:C_6$, $C_2^4\times C_{24}$, $C_{12}.C_2^5$, $C_2^2\times D_4:C_{12}$

Order 360: $C_3\times \SL(2,5)$

Order 320: $D_{10}:C_4^2$ x 198, $C_2^4\times F_5$ x 60, $C_{20}:C_2^4$ x 28, $C_2^4\times D_{10}$, $C_2^4\times C_{20}$, $C_{10}.C_2^5$

Order 288: $C_4\times S_3\times A_4$ x 16, $C_{24}:D_6$ x 12, $C_2\times A_4\times D_6$ x 10, $Q_8.C_6^2$ x 6, $C_2^3.C_6^2$ x 6, $C_6:C_4\times A_4$ x 6, $A_4\times C_{24}$ x 4, $A_4\times C_3:C_8$ x 4, $\SL(2,3):C_{12}$ x 4, $C_6^2.C_2^3$, $Q_8:C_6^2$, $A_4\times \SL(2,3)$, $C_2\times C_6\times C_{24}$, $C_6^2:C_8$, $C_2^3:C_6^2$

Order 256: $C_4^2:C_2^4$ x 28, $D_4:C_2^5$, $C_2^4\times \OD_{16}$, $C_2^4\times C_4^2$

Order 240: $D_{10}:C_{12}$ x 28, $A_4\times D_{10}$ x 28, $A_4\times F_5$ x 16, $C_{12}\times F_5$ x 16, $C_{12}\times D_{10}$ x 12, $\SL(2,5):C_2$ x 8, $C_2\times \SL(2,5)$ x 7, $C_2^3:C_{30}$ x 7, $A_4\times C_{20}$ x 4, $A_4\times C_5:C_4$ x 4, $C_{15}:C_2^4$, $C_2^2\times C_{60}$, $C_{30}.C_2^3$

Order 192: $C_{24}:C_2^3$ x 198, $\SL(2,3):C_2^3$ x 52, $C_2\times \Unitary(2,3)$ x 44, $C_2^4:C_{12}$ x 29, $C_{12}:C_2^4$ x 28, $A_4\times D_4:C_2$ x 20, $C_2^3\times C_{24}$ x 14, $C_{12}.C_2^4$ x 14, $C_4\wr C_2\times C_6$ x 12, $C_2^3:C_{24}$ x 12, $A_4\times C_4^2$ x 10, $A_4\times \OD_{16}$ x 10, $C_2^5:C_6$ x 9, $C_2^3\times \SL(2,3)$ x 7, $C_2\times Q_8\times A_4$ x 6, $Q_8.(C_2\times A_4)$ x 4, $C_2^3:\SL(2,3)$ x 3, $D_6\times C_2^4$, $A_4\times C_2^4$, $C_{12}.C_2^4$, $C_2^4\times C_{12}$, $C_6.C_2^5$, $C_{24}:C_2^3$, $C_2^2\times C_4\times C_{12}$

Order 160: $C_2^3\times F_5$ x 452, $C_{10}:C_4^2$ x 328, $C_{20}:C_2^3$ x 198, $C_2^3\times D_{10}$ x 30, $C_2^3\times C_{20}$ x 14, $C_{10}.C_2^4$ x 14, $C_2^4\times C_{10}$

Order 144: $C_{24}:C_6$ x 16, $A_4\times D_6$ x 16, $C_{12}\times D_6$ x 12, $C_2^2:C_6^2$ x 7, $C_6:C_{24}$ x 6, $C_6\times C_{24}$ x 6, $\SL(2,3):C_6$ x 4, $C_{12}\times A_4$ x 4, $A_4\times C_3:C_4$ x 4, $C_6\times \SL(2,3)$ x 3, $C_6^2:C_2^2$, $C_2\times C_6\times C_{12}$, $C_6^2:C_4$

Order 128: $C_4^2:C_2^3$ x 198, $D_4:C_2^4$ x 28, $C_2^3\times \OD_{16}$ x 21, $C_2^3\times C_4^2$ x 21, $C_2^5\times C_4$ x 2, $Q_8\times C_2^4$, $D_4\times C_2^4$, $C_2^4\times C_8$

Order 120: $C_6\times F_5$ x 56, $D_5\times C_{12}$ x 16, $C_6\times D_{10}$ x 14, $D_5\times A_4$ x 8, $C_{10}\times A_4$ x 7, $C_2\times C_{60}$ x 6, $C_{10}:C_{12}$ x 6, $\SL(2,5)$, $C_2^2\times C_{30}$

Order 96: $C_8:D_6$ x 328, $C_{12}:C_2^3$ x 198, $C_2^3:C_{12}$ x 62, $\SL(2,3):C_2^2$ x 56, $C_2^2\times C_{24}$ x 52, $C_{12}.C_2^3$ x 50, $C_2^3\times D_6$ x 22, $D_4:C_{12}$ x 16, $C_2^3\times C_{12}$ x 16, $C_2^2\times \SL(2,3)$ x 15, $C_6.C_2^4$ x 14, $C_2^3\times A_4$ x 12, $C_6.C_2^4$ x 12, $D_4\times A_4$ x 10, $C_6\times \OD_{16}$ x 9, $C_2\times C_4\times C_{12}$ x 9, $C_8\times A_4$ x 8, $\Unitary(2,3)$ x 8, $Q_8\times A_4$ x 5, $C_2^4\times C_6$, $C_6.C_2^4$, $C_{12}:C_2^3$, $Q_8:A_4$

Order 80: $C_2^2\times F_5$ x 856, $C_4\times D_{10}$ x 328, $C_2^2\times D_{10}$ x 226, $C_4\times F_5$ x 96, $C_2^2\times C_{20}$ x 50, $C_2^2.D_{10}$ x 50, $C_2^3\times C_{10}$ x 15

Order 72: $S_3\times C_{12}$ x 16, $C_6\times D_6$ x 10, $C_6\times A_4$ x 7, $C_6\times C_{12}$ x 6, $C_6:C_{12}$ x 6, $S_3\times A_4$ x 4, $C_3\times C_{24}$ x 4, $C_3:C_{24}$ x 4, $C_2\times C_6^2$, $C_3\times \SL(2,3)$

Order 64: $C_4^2:C_2^2$ x 328, $D_4:C_2^3$ x 248, $C_2^2\times \OD_{16}$ x 124, $C_2^2\times C_4^2$ x 124, $C_2^4\times C_4$ x 60, $D_4\times C_2^3$ x 21, $Q_8\times C_2^3$ x 14, $C_2^3\times C_8$ x 14, $C_2^6$

Order 60: $C_3\times D_{10}$ x 28, $C_3\times F_5$ x 16, $C_2\times C_{30}$ x 7, $C_{60}$ x 4, $C_5:C_{12}$ x 4, $C_5\times A_4$

Order 48: $C_4\times D_6$ x 328, $C_2^2\times D_6$ x 138, $C_{24}:C_2$ x 96, $C_2^2\times C_{12}$ x 79, $C_2\times C_{24}$ x 56, $C_6.C_2^3$ x 50, $C_6:C_8$ x 44, $C_2^2\times A_4$ x 30, $D_4:C_6$ x 20, $C_4\times A_4$ x 20, $C_2^3\times C_6$ x 16, $\SL(2,3):C_2$ x 16, $C_2\times \SL(2,3)$ x 13, $C_3\times \OD_{16}$ x 10, $C_4\times C_{12}$ x 10, $C_6\times D_4$ x 9, $C_6\times Q_8$ x 6

Order 40: $C_2\times D_{10}$ x 428, $C_2\times F_5$ x 368, $C_4\times D_5$ x 96, $C_2^2\times C_{10}$ x 57, $C_2\times C_{20}$ x 44, $C_{10}:C_4$ x 44

Order 36: $C_6\times S_3$ x 16, $C_6^2$ x 7, $C_3\times C_{12}$ x 4, $C_3:C_{12}$ x 4, $C_3\times A_4$

Order 32: $D_4:C_2^2$ x 612, $C_2^3\times C_4$ x 452, $C_2\times \OD_{16}$ x 186, $C_2\times C_4^2$ x 186, $C_2^2\times D_4$ x 124, $C_4\wr C_2$ x 96, $C_2^2\times Q_8$ x 63, $C_2^2\times C_8$ x 50, $C_2^5$ x 23

Order 30: $C_3\times D_5$ x 8, $C_{30}$ x 7

Order 24: $C_2\times D_6$ x 236, $C_2\times C_{12}$ x 106, $C_4\times S_3$ x 96, $C_2^2\times C_6$ x 69, $C_6:C_4$ x 44, $C_2\times A_4$ x 18, $C_{24}$ x 16, $C_3\times D_4$ x 10, $C_3:C_8$ x 8, $C_3\times Q_8$ x 5, $\SL(2,3)$ x 3

Order 20: $D_{10}$ x 184, $C_2\times C_{10}$ x 57, $F_5$ x 32, $C_{20}$ x 8, $C_5:C_4$ x 8

Order 18: $C_3\times C_6$ x 7, $C_3\times S_3$ x 4

Order 16: $C_2^2\times C_4$ x 856, $D_4:C_2$ x 288, $C_2\times D_4$ x 186, $C_2^4$ x 153, $C_2\times Q_8$ x 81, $\OD_{16}$ x 52, $C_4^2$ x 52, $C_2\times C_8$ x 44

Order 15: $C_{15}$

Order 12: $D_6$ x 96, $C_2\times C_6$ x 87, $C_{12}$ x 28, $C_3:C_4$ x 8, $A_4$ x 2

Order 10: $D_5$ x 16, $C_{10}$ x 15

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 368, $C_2^3$ x 293, $D_4$ x 52, $Q_8$ x 21, $C_8$ x 8

Order 6: $C_6$ x 33, $S_3$ x 8

Order 5: $C_5$

Order 4: $C_2^2$ x 153, $C_4$ x 32

Order 3: $C_3$ x 3

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 23040: $C_2^2\times A_4\times \GL(2,5)$

Order 11520: $A_4\times \SL(2,5):C_2^3$, $C_2\times A_4\times \GL(2,5)$

Order 7680: $C_2^4\times \GL(2,5)$

Order 5760: $A_4\times \SL(2,5):C_2^2$ x 2, $C_2\times C_6\times \GL(2,5)$, $A_4\times \GL(2,5)$, $C_2^5.\GL(2,4)$

Order 4608: $(C_2^4\times \SL(2,3)):C_{12}$

Order 3840: $C_2^3\times \GL(2,5)$ x 4, $\SL(2,5):C_2^5$, $A_4\times D_{10}:C_4^2$

Order 2880: $A_4\times \SL(2,5):C_2$ x 2, $C_2^2\times \SL(2,5):C_6$, $C_2^4.\GL(2,4)$, $C_6\times \GL(2,5)$

Order 2304: $A_4\times C_{24}:C_2^3$, $A_4\times \SL(2,3):C_2^3$, $(C_2^3\times \SL(2,3)):C_{12}$

Order 1920: $C_2^2\times \GL(2,5)$ x 9, $\SL(2,5):C_2^4$ x 6, $A_4\times C_{10}:C_4^2$ x 2, $C_2^4\times \SL(2,5)$, $C_2^3\times A_4\times F_5$, $C_2^3:C_{12}\times D_{10}$

Order 1536: $C_2^4\times \Unitary(2,3)$, $A_4\times C_4^2:C_2^3$

Order 1440: $C_2\times \SL(2,5):C_6$ x 2, $C_2^3.\GL(2,4)$, $A_4\times \SL(2,5)$, $C_3\times \GL(2,5)$

Order 1280: $D_{10}.C_2^6$

Order 1152: $A_4\times \SL(2,3):C_2^2$ x 2, $C_8:D_6\times A_4$ x 2, $C_2^3:C_{12}\times D_6$, $C_2^3.A_4^2$, $(C_2^3\times C_6):C_{24}$, $C_2^2\times \SL(2,3):C_{12}$, $A_4\times \Unitary(2,3)$, $C_2^2\times A_4\times C_{24}$

Order 960: $\SL(2,5):C_2^3$ x 16, $C_2\times \GL(2,5)$ x 6, $C_4\times A_4\times D_{10}$ x 4, $C_2^3\times \SL(2,5)$ x 3, $C_2^2\times A_4\times F_5$ x 2, $C_4\times A_4\times F_5$ x 2, $C_2^2\times A_4\times D_{10}$, $C_2^4:C_{60}$, $C_2^2\times C_{12}\times F_5$, $(C_2^3\times C_{10}):C_{12}$

Order 768: $C_2^3\times \Unitary(2,3)$ x 4, $A_4\times C_4^2:C_2^2$ x 2, $Q_8.(C_2^3\times A_4)$, $\SL(2,3):C_2^5$, $D_4:C_2^3\times A_4$, $C_2^6.C_{12}$, $C_2^2\times A_4\times C_4^2$, $C_{24}:C_2^5$

Order 720: $\SL(2,5):C_6$ x 2, $C_6\times \SL(2,5)$

Order 640: $D_{10}.C_2^5$ x 7, $F_5\times C_2^5$, $C_{20}:C_2^5$

Order 576: $C_4\times A_4\times D_6$ x 4, $C_4.A_4^2$ x 2, $A_4\times C_{24}:C_2$ x 2, $C_2^2\times A_4\times D_6$, $C_2^2\times \SL(2,3):C_6$, $C_2^4.C_6^2$, $C_6.C_2^3\times A_4$, $C_2^2.A_4^2$, $C_6^2:\OD_{16}$, $C_2\times A_4\times C_{24}$, $C_6:C_8\times A_4$, $C_6\times \Unitary(2,3)$

Order 512: $C_4^2:C_2^5$

Order 480: $\SL(2,5):C_2^2$ x 13, $C_2^2\times \SL(2,5)$ x 6, $C_4\times D_5\times A_4$ x 5, $C_2\times A_4\times D_{10}$ x 3, $\GL(2,5)$ x 2, $C_2\times A_4\times F_5$ x 2, $C_2^3:C_{60}$ x 2, $C_{30}:C_4^2$ x 2, $C_{10}:C_4\times A_4$ x 2, $C_2^4:C_{30}$, $C_2^2\times C_6\times F_5$, $C_{60}:C_2^3$

Order 384: $C_2^2\times \Unitary(2,3)$ x 9, $C_{24}:C_2^4$ x 8, $\SL(2,3):C_2^4$ x 6, $D_4:C_2^2\times A_4$ x 4, $A_4\times C_4\wr C_2$ x 2, $C_2^4.(C_2\times A_4)$ x 2, $C_2^5:C_{12}$ x 2, $C_2^5.C_{12}$ x 2, $C_2^4:C_{24}$ x 2, $C_2\times A_4\times C_4^2$ x 2, $C_{12}:C_2^5$, $C_2^4:\SL(2,3)$, $C_2^2\times Q_8\times A_4$, $C_2^4\times \SL(2,3)$, $C_2^6:C_6$, $C_2^4\times C_{24}$, $C_{12}.C_2^5$, $C_2^2\times D_4:C_{12}$

Order 360: $C_3\times \SL(2,5)$

Order 320: $D_{10}:C_4^2$ x 22, $C_{20}:C_2^4$ x 12, $C_2^4\times F_5$ x 6, $C_2^4\times D_{10}$, $C_2^4\times C_{20}$, $C_{10}.C_2^5$

Order 288: $C_4\times S_3\times A_4$ x 5, $C_2\times A_4\times D_6$ x 3, $Q_8.C_6^2$ x 2, $C_2^3.C_6^2$ x 2, $C_6:C_4\times A_4$ x 2, $C_{24}:D_6$ x 2, $C_6^2.C_2^3$, $Q_8:C_6^2$, $A_4\times \SL(2,3)$, $C_2\times C_6\times C_{24}$, $C_6^2:C_8$, $A_4\times C_{24}$, $A_4\times C_3:C_8$, $\SL(2,3):C_{12}$, $C_2^3:C_6^2$

Order 256: $C_4^2:C_2^4$ x 8, $D_4:C_2^5$, $C_2^4\times \OD_{16}$, $C_2^4\times C_4^2$

Order 240: $\SL(2,5):C_2$ x 4, $A_4\times D_{10}$ x 4, $C_{12}\times D_{10}$ x 4, $C_2\times \SL(2,5)$ x 3, $C_2^3:C_{30}$ x 2, $D_{10}:C_{12}$ x 2, $A_4\times C_{20}$ x 2, $C_{12}\times F_5$ x 2, $A_4\times C_5:C_4$ x 2, $C_{15}:C_2^4$, $A_4\times F_5$, $C_2^2\times C_{60}$, $C_{30}.C_2^3$

Order 192: $C_{24}:C_2^3$ x 26, $\SL(2,3):C_2^3$ x 18, $C_{12}:C_2^4$ x 12, $C_2^4:C_{12}$ x 9, $A_4\times D_4:C_2$ x 7, $C_2\times \Unitary(2,3)$ x 6, $C_2^3\times C_{24}$ x 4, $C_{12}.C_2^4$ x 4, $C_2^3\times \SL(2,3)$ x 3, $C_2^5:C_6$ x 3, $A_4\times C_4^2$ x 2, $C_4\wr C_2\times C_6$ x 2, $Q_8.(C_2\times A_4)$ x 2, $C_2\times Q_8\times A_4$ x 2, $A_4\times \OD_{16}$ x 2, $C_2^3:C_{24}$ x 2, $D_6\times C_2^4$, $A_4\times C_2^4$, $C_{12}.C_2^4$, $C_2^4\times C_{12}$, $C_6.C_2^5$, $C_2^3:\SL(2,3)$, $C_{24}:C_2^3$, $C_2^2\times C_4\times C_{12}$

Order 160: $C_{20}:C_2^3$ x 51, $C_{10}:C_4^2$ x 22, $C_2^3\times F_5$ x 17, $C_2^3\times D_{10}$ x 8, $C_2^3\times C_{20}$ x 6, $C_{10}.C_2^4$ x 6, $C_2^4\times C_{10}$

Order 144: $A_4\times D_6$ x 4, $C_{12}\times D_6$ x 4, $C_{24}:C_6$ x 2, $C_2^2:C_6^2$ x 2, $\SL(2,3):C_6$ x 2, $C_{12}\times A_4$ x 2, $A_4\times C_3:C_4$ x 2, $C_6:C_{24}$, $C_6^2:C_2^2$, $C_2\times C_6\times C_{12}$, $C_6^2:C_4$, $C_6\times \SL(2,3)$, $C_6\times C_{24}$

Order 128: $C_4^2:C_2^3$ x 26, $D_4:C_2^4$ x 12, $C_2^3\times \OD_{16}$ x 7, $C_2^3\times C_4^2$ x 7, $C_2^5\times C_4$ x 2, $Q_8\times C_2^4$, $D_4\times C_2^4$, $C_2^4\times C_8$

Order 120: $D_5\times C_{12}$ x 5, $C_6\times D_{10}$ x 3, $C_{10}\times A_4$ x 2, $C_6\times F_5$ x 2, $D_5\times A_4$ x 2, $C_2\times C_{60}$ x 2, $C_{10}:C_{12}$ x 2, $\SL(2,5)$, $C_2^2\times C_{30}$

Order 96: $C_{12}:C_2^3$ x 51, $C_8:D_6$ x 28, $\SL(2,3):C_2^2$ x 17, $C_2^3:C_{12}$ x 13, $C_2^2\times C_{24}$ x 11, $C_{12}.C_2^3$ x 9, $C_2^3\times D_6$ x 8, $C_2^3\times C_{12}$ x 8, $C_2^2\times \SL(2,3)$ x 8, $C_6.C_2^4$ x 6, $C_2^3\times A_4$ x 5, $C_6.C_2^4$ x 4, $D_4\times A_4$ x 4, $Q_8\times A_4$ x 3, $C_8\times A_4$ x 2, $\Unitary(2,3)$ x 2, $D_4:C_{12}$ x 2, $C_6\times \OD_{16}$ x 2, $C_2\times C_4\times C_{12}$ x 2, $C_2^4\times C_6$, $C_6.C_2^4$, $C_{12}:C_2^3$, $Q_8:A_4$

Order 80: $C_4\times D_{10}$ x 68, $C_2^2\times D_{10}$ x 28, $C_2^2\times F_5$ x 19, $C_2^2\times C_{20}$ x 16, $C_2^2.D_{10}$ x 16, $C_4\times F_5$ x 8, $C_2^3\times C_{10}$ x 5

Order 72: $S_3\times C_{12}$ x 5, $C_6\times D_6$ x 3, $C_6\times A_4$ x 2, $S_3\times A_4$ x 2, $C_6\times C_{12}$ x 2, $C_6:C_{12}$ x 2, $C_2\times C_6^2$, $C_3\times \SL(2,3)$, $C_3\times C_{24}$, $C_3:C_{24}$

Order 64: $D_4:C_2^3$ x 67, $C_4^2:C_2^2$ x 28, $C_2^2\times \OD_{16}$ x 22, $C_2^2\times C_4^2$ x 22, $C_2^4\times C_4$ x 20, $D_4\times C_2^3$ x 9, $Q_8\times C_2^3$ x 6, $C_2^3\times C_8$ x 4, $C_2^6$

Order 60: $C_3\times D_{10}$ x 4, $C_{60}$ x 2, $C_5:C_{12}$ x 2, $C_2\times C_{30}$ x 2, $C_5\times A_4$, $C_3\times F_5$

Order 48: $C_4\times D_6$ x 68, $C_2^2\times D_6$ x 28, $C_2^2\times C_{12}$ x 25, $C_6.C_2^3$ x 16, $C_{24}:C_2$ x 10, $C_2^2\times A_4$ x 8, $\SL(2,3):C_2$ x 8, $C_2\times C_{24}$ x 8, $D_4:C_6$ x 7, $C_4\times A_4$ x 7, $C_6:C_8$ x 6, $C_2^3\times C_6$ x 6, $C_2\times \SL(2,3)$ x 5, $C_6\times D_4$ x 3, $C_6\times Q_8$ x 2, $C_3\times \OD_{16}$ x 2, $C_4\times C_{12}$ x 2

Order 40: $C_2\times D_{10}$ x 38, $C_4\times D_5$ x 25, $C_2\times C_{20}$ x 13, $C_{10}:C_4$ x 13, $C_2^2\times C_{10}$ x 12, $C_2\times F_5$ x 10

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

Order 32: $D_4:C_2^2$ x 123, $C_2^3\times C_4$ x 80, $C_2^2\times D_4$ x 36, $C_2^2\times Q_8$ x 22, $C_2\times \OD_{16}$ x 22, $C_2\times C_4^2$ x 22, $C_4\wr C_2$ x 10, $C_2^5$ x 9, $C_2^2\times C_8$ x 9

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

Order 24: $C_2\times D_6$ x 38, $C_2\times C_{12}$ x 26, $C_4\times S_3$ x 25, $C_2^2\times C_6$ x 17, $C_6:C_4$ x 13, $C_2\times A_4$ x 6, $C_{24}$ x 4, $C_3\times D_4$ x 4, $\SL(2,3)$ x 3, $C_3\times Q_8$ x 3, $C_3:C_8$ x 2

Order 20: $D_{10}$ x 21, $C_2\times C_{10}$ x 12, $C_{20}$ x 4, $C_5:C_4$ x 4, $F_5$ x 2

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

Order 16: $C_2^2\times C_4$ x 119, $D_4:C_2$ x 65, $C_2\times D_4$ x 44, $C_2^4$ x 33, $C_2\times Q_8$ x 23, $\OD_{16}$ x 8, $C_4^2$ x 8, $C_2\times C_8$ x 6

Order 15: $C_{15}$

Order 12: $D_6$ x 21, $C_2\times C_6$ x 20, $C_{12}$ x 11, $C_3:C_4$ x 4, $A_4$ x 2

Order 10: $C_{10}$ x 5, $D_5$ x 4

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 61, $C_2^3$ x 50, $D_4$ x 17, $Q_8$ x 10, $C_8$ x 2

Order 6: $C_6$ x 11, $S_3$ x 4

Order 5: $C_5$

Order 4: $C_2^2$ x 33, $C_4$ x 10

Order 3: $C_3$ x 3

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 23040: $C_2^2\times A_4\times \GL(2,5)$ ($C_1$)

Order 11520: $C_2\times A_4\times \GL(2,5)$ ($C_2$) x 6, $A_4\times \SL(2,5):C_2^3$ ($C_2$)

Order 7680: $C_2^4\times \GL(2,5)$ ($C_3$)

Order 5760: $A_4\times \GL(2,5)$ ($C_2^2$) x 4, $A_4\times \SL(2,5):C_2^2$ ($C_2^2$) x 3, $A_4\times \SL(2,5):C_2^2$ ($C_4$) x 3, $C_2^5.\GL(2,4)$ ($C_4$)

Order 3840: $C_2^3\times \GL(2,5)$ ($C_6$) x 6, $\SL(2,5):C_2^5$ ($C_6$)

Order 2880: $A_4\times \SL(2,5):C_2$ ($C_2\times C_4$) x 3, $C_2^4.\GL(2,4)$ ($C_2\times C_4$) x 3, $A_4\times \SL(2,5):C_2$ ($C_2^3$)

Order 1920: $C_2^2\times \GL(2,5)$ ($C_2\times C_6$) x 4, $\SL(2,5):C_2^4$ ($C_2\times C_6$) x 3, $\SL(2,5):C_2^4$ ($C_{12}$) x 3, $C_2^4\times \SL(2,5)$ ($C_{12}$), $C_2^2\times \GL(2,5)$ ($A_4$)

Order 1440: $A_4\times \SL(2,5)$ ($C_2^2\times C_4$)

Order 960: $C_2\times \GL(2,5)$ ($C_2\times A_4$) x 6, $C_2^3\times \SL(2,5)$ ($C_2\times C_{12}$) x 3, $\SL(2,5):C_2^3$ ($C_2\times C_{12}$) x 3, $\SL(2,5):C_2^3$ ($C_2^2\times C_6$), $\SL(2,5):C_2^3$ ($C_2\times A_4$)

Order 480: $\GL(2,5)$ ($C_2^2\times A_4$) x 4, $\SL(2,5):C_2^2$ ($C_2^2\times A_4$) x 3, $\SL(2,5):C_2^2$ ($C_4\times A_4$) x 3, $C_2^2\times \SL(2,5)$ ($C_2^2\times C_{12}$), $C_2^2\times \SL(2,5)$ ($C_4\times A_4$)

Order 240: $C_2\times \SL(2,5)$ ($C_2^3:C_{12}$) x 3, $\SL(2,5):C_2$ ($C_2^3:C_{12}$) x 3, $\SL(2,5):C_2$ ($C_2^3\times A_4$)

Order 192: $C_2^4:C_{12}$ ($S_5$)

Order 120: $\SL(2,5)$ ($C_2^4:C_{12}$)

Order 96: $C_2^3:C_{12}$ ($A_5:C_4$) x 3, $C_2^3:C_{12}$ ($C_2\times S_5$) x 3, $C_2^3\times A_4$ ($A_5:C_4$)

Order 64: $C_2^4\times C_4$ ($C_3\times S_5$)

Order 48: $C_2^2\times A_4$ ($\GL(2,5)$) x 4, $C_2^2\times A_4$ ($C_2^2.S_5$) x 3, $C_4\times A_4$ ($C_2^2.S_5$) x 3, $C_4\times A_4$ ($C_2^2\times S_5$)

Order 32: $C_2^3\times C_4$ ($C_6\times S_5$) x 3, $C_2^3\times C_4$ ($A_5:C_{12}$) x 3, $C_2^5$ ($A_5:C_{12}$)

Order 24: $C_2\times A_4$ ($C_2\times \GL(2,5)$) x 6, $C_2\times A_4$ ($C_2^3.S_5$)

Order 16: $C_2^4$ ($C_3\times \GL(2,5)$) x 4, $C_2^4$ ($A_5:C_4\times C_6$) x 3, $C_2^2\times C_4$ ($A_5:C_4\times C_6$) x 3, $C_2^2\times C_4$ ($\GL(2,4):C_2^3$), $C_2^2\times C_4$ ($A_4\times S_5$)

Order 12: $A_4$ ($C_2^2\times \GL(2,5)$)

Order 8: $C_2^3$ ($C_6\times \GL(2,5)$) x 6, $C_2\times C_4$ ($A_4\times A_5:C_4$) x 3, $C_2\times C_4$ ($C_2\times A_4\times S_5$) x 3, $C_2^3$ ($(C_2^2\times C_6).S_5$), $C_2^3$ ($A_4\times A_5:C_4$)

Order 4: $C_2^2$ ($A_4\times \GL(2,5)$) x 4, $C_2^2$ ($A_4\times C_2^2.S_5$) x 3, $C_4$ ($A_4\times C_2^2.S_5$) x 3, $C_2^2$ ($C_2\times C_6\times \GL(2,5)$), $C_4$ ($C_2^2\times A_4\times S_5$)

Order 2: $C_2$ ($C_2\times A_4\times \GL(2,5)$) x 6, $C_2$ ($(C_2^3\times A_4).S_5$)

Order 1: $C_1$ ($C_2^2\times A_4\times \GL(2,5)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 23040: $C_2^2\times A_4\times \GL(2,5)$ ($C_1$)

Order 11520: $A_4\times \SL(2,5):C_2^3$ ($C_2$), $C_2\times A_4\times \GL(2,5)$ ($C_2$)

Order 7680: $C_2^4\times \GL(2,5)$ ($C_3$)

Order 5760: $A_4\times \GL(2,5)$ ($C_2^2$), $C_2^5.\GL(2,4)$ ($C_4$), $A_4\times \SL(2,5):C_2^2$ ($C_2^2$), $A_4\times \SL(2,5):C_2^2$ ($C_4$)

Order 3840: $\SL(2,5):C_2^5$ ($C_6$), $C_2^3\times \GL(2,5)$ ($C_6$)

Order 2880: $A_4\times \SL(2,5):C_2$ ($C_2^3$), $A_4\times \SL(2,5):C_2$ ($C_2\times C_4$), $C_2^4.\GL(2,4)$ ($C_2\times C_4$)

Order 1920: $\SL(2,5):C_2^4$ ($C_2\times C_6$), $\SL(2,5):C_2^4$ ($C_{12}$), $C_2^4\times \SL(2,5)$ ($C_{12}$), $C_2^2\times \GL(2,5)$ ($C_2\times C_6$), $C_2^2\times \GL(2,5)$ ($A_4$)

Order 1440: $A_4\times \SL(2,5)$ ($C_2^2\times C_4$)

Order 960: $C_2\times \GL(2,5)$ ($C_2\times A_4$), $C_2^3\times \SL(2,5)$ ($C_2\times C_{12}$), $\SL(2,5):C_2^3$ ($C_2\times C_{12}$), $\SL(2,5):C_2^3$ ($C_2^2\times C_6$), $\SL(2,5):C_2^3$ ($C_2\times A_4$)

Order 480: $C_2^2\times \SL(2,5)$ ($C_2^2\times C_{12}$), $C_2^2\times \SL(2,5)$ ($C_4\times A_4$), $\SL(2,5):C_2^2$ ($C_2^2\times A_4$), $\SL(2,5):C_2^2$ ($C_4\times A_4$), $\GL(2,5)$ ($C_2^2\times A_4$)

Order 240: $C_2\times \SL(2,5)$ ($C_2^3:C_{12}$), $\SL(2,5):C_2$ ($C_2^3\times A_4$), $\SL(2,5):C_2$ ($C_2^3:C_{12}$)

Order 192: $C_2^4:C_{12}$ ($S_5$)

Order 120: $\SL(2,5)$ ($C_2^4:C_{12}$)

Order 96: $C_2^3\times A_4$ ($A_5:C_4$), $C_2^3:C_{12}$ ($A_5:C_4$), $C_2^3:C_{12}$ ($C_2\times S_5$)

Order 64: $C_2^4\times C_4$ ($C_3\times S_5$)

Order 48: $C_2^2\times A_4$ ($C_2^2.S_5$), $C_2^2\times A_4$ ($\GL(2,5)$), $C_4\times A_4$ ($C_2^2.S_5$), $C_4\times A_4$ ($C_2^2\times S_5$)

Order 32: $C_2^5$ ($A_5:C_{12}$), $C_2^3\times C_4$ ($C_6\times S_5$), $C_2^3\times C_4$ ($A_5:C_{12}$)

Order 24: $C_2\times A_4$ ($C_2\times \GL(2,5)$), $C_2\times A_4$ ($C_2^3.S_5$)

Order 16: $C_2^4$ ($A_5:C_4\times C_6$), $C_2^4$ ($C_3\times \GL(2,5)$), $C_2^2\times C_4$ ($\GL(2,4):C_2^3$), $C_2^2\times C_4$ ($A_4\times S_5$), $C_2^2\times C_4$ ($A_5:C_4\times C_6$)

Order 12: $A_4$ ($C_2^2\times \GL(2,5)$)

Order 8: $C_2^3$ ($(C_2^2\times C_6).S_5$), $C_2^3$ ($C_6\times \GL(2,5)$), $C_2^3$ ($A_4\times A_5:C_4$), $C_2\times C_4$ ($A_4\times A_5:C_4$), $C_2\times C_4$ ($C_2\times A_4\times S_5$)

Order 4: $C_2^2$ ($C_2\times C_6\times \GL(2,5)$), $C_2^2$ ($A_4\times \GL(2,5)$), $C_2^2$ ($A_4\times C_2^2.S_5$), $C_4$ ($A_4\times C_2^2.S_5$), $C_4$ ($C_2^2\times A_4\times S_5$)

Order 2: $C_2$ ($(C_2^3\times A_4).S_5$), $C_2$ ($C_2\times A_4\times \GL(2,5)$)

Order 1: $C_1$ ($C_2^2\times A_4\times \GL(2,5)$)

Series

Derived series

$C_2^2\times A_4\times \GL(2,5)$

$\rhd$

$C_2^2\times \SL(2,5)$

$\rhd$

$\SL(2,5)$

Chief series

$C_2^2\times A_4\times \GL(2,5)$

$\rhd$

$A_4\times \SL(2,5):C_2^3$

$\rhd$

$C_2^4:C_{12}$

$\rhd$

$C_2^3\times A_4$

$\rhd$

$C_2^2\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^2\times A_4\times \GL(2,5)$

$\rhd$

$C_2^2\times \SL(2,5)$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_4$

Supergroups

This group is a maximal subgroup of 1 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 $384 \times 384$ 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 $180 \times 180$ rational character table (warning: may be slow to load).