Abstract group 32440320.a: $C_2^{12}.M_{11}$

• Label: 32440320.a
• Order: \( 2^{16} \cdot 3^{2} \cdot 5 \cdot 11 \)
• Exponent: \( 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3^{2} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $24$
• Trans deg.: $24$
• Rank: $2$

Group information

Description:	$C_2^{12}.M_{11}$
Order:	\(32440320\)\(\medspace = 2^{16} \cdot 3^{2} \cdot 5 \cdot 11 \)
Exponent:	\(2640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	$C_2^{12}.M_{11}$, of order \(32440320\)\(\medspace = 2^{16} \cdot 3^{2} \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 12, $M_{11}$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	10	11	12	16	22
Elements	1	46335	28160	1647360	405504	3125760	7096320	6082560	1474560	4055040	4055040	4423680	32440320
Conjugacy classes	1	23	1	33	1	27	11	15	2	12	4	6	136
Divisions	1	23	1	33	1	27	9	9	1	12	2	3	122
Autjugacy classes	1	18	1	26	1	20	11	11	2	9	4	4	108

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$24$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $24$ $\langle(3,17,11,20,23,4,18,12,19,24)(5,10,21,13,15)(6,9,22,14,16), (1,24,10,7,5,13)(2,23,9,8,6,14)(11,16,12,15)(17,21,20)(18,22,19)\rangle$
Transitive group:	24T24338				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$C_2^{12}$ . $M_{11}$	$(C_2^{11}.M_{11})$ . $C_2$	$C_2^{11}$ . $(C_2\times M_{11})$	$C_2$ . $(C_2^{11}.M_{11})$	more information

Elements of the group are displayed as permutations of degree 24.

Homology

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

Subgroups

There are 6 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^{11}.M_{11}$
Commutator:	$G' \simeq$ $C_2^{11}.M_{11}$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_2^{11}$	$G/\Phi \simeq$ $C_2\times M_{11}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^{12}$	$G/\operatorname{Fit} \simeq$ $M_{11}$
Radical:	$R \simeq$ $C_2^{12}$	$G/R \simeq$ $M_{11}$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_2^{11}.M_{11}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6.C_2^5.C_2^4.C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 2112 (order at least 15360) are shown, as well as all normal subgroups of any index.

Order 32440320: $C_2^{12}.M_{11}$

Order 16220160: $C_2^{11}.M_{11}$

Order 2949120: $C_2^{10}.A_6.D_4$

Order 2703360: $C_2^{12}.\PSL(2,11)$

Order 1474560: $C_2^{12}.A_6$, $C_2^{11}.A_6.C_2$, $C_2^{10}.A_6:C_4$

Order 1351680: $C_2^{11}.\PSL(2,11)$ x 3

Order 737280: $C_2^{11}.A_6$, $C_2^{11}.A_6$, $C_2^{10}.A_6.C_2$

Order 675840: $C_2^{10}.\PSL(2,11)$

Order 589824: $C_2^{12}.F_9:C_2$

Order 491520: $C_2^{12}.S_5$

Order 368640: $C_2^{10}.A_6$

Order 294912: $C_2^{10}.F_9:C_2^2$ x 2, $C_2^9.(C_6^2.\SD_{16})$ x 2, $C_2^8.C_6^2.C_2^4.C_2$, $C_2^{12}.\PSU(3,2)$, $C_2^{12}.F_9$

Order 245760: $C_2^{12}.A_5$ x 2, $C_2^{11}.S_5$ x 2, $C_2^{11}.S_5$ x 2, $C_2^{11}.S_5$, $C_2^{11}.S_5$

Order 225280: $C_2^{12}:C_{11}:C_5$

Order 196608: $C_2^5.C_2^6.A_4.C_2^3$

Order 147456: $C_2^8.C_6^2.C_2^3.C_2$ x 3, $C_2^8.C_3^2.C_2^4.C_2^2$ x 2, $C_2^8.C_6^2.C_2^3.C_2$ x 2, $C_2^{10}.F_9:C_2$ x 2, $C_2^{10}.F_9:C_2$ x 2, $C_2^9.(C_6^2.Q_8)$ x 2, $C_2^{10}.(C_2\times F_9)$ x 2, $C_2^8.C_3^2.C_2^5.C_2$, $C_2^8.(C_2\times D_6:D_6).C_2$, $C_2^8.C_6^2.C_2^4$, $C_2^9.C_6^2:Q_8$

Order 122880: $C_2^{11}.A_5$ x 10, $C_2^{11}.A_5$ x 2, $C_2^{10}.S_5$ x 2, $C_2^{10}.S_5$ x 2, $C_2^{10}.S_5$ x 2

Order 112640: $C_2^{10}.C_{11}:C_{10}$ x 3

Order 98304: $C_2^6.C_2.C_2^6.D_6$ x 4, $C_2^6.C_2.C_2^6.D_6$, $C_2^6.C_2^5.A_4.C_2^2$, $C_2^8.A_4.C_2^4.C_2$, $C_2^6.C_2.C_2^6.D_6$

Order 81920: $C_2^{12}.F_5$

Order 73728: $C_2^8.C_6^2.C_2^3$ x 14, $C_2^8.C_3^2.C_2^4.C_2$ x 8, $C_2^{10}.\SOPlus(4,2)$ x 8, $C_2^{10}.\PSU(3,2)$ x 4, $C_2^8.C_3:S_3.C_4^2$ x 3, $C_2^{10}.\SOPlus(4,2)$ x 3, $C_2^8.D_6^2.C_2$ x 2, $C_2^8.(C_2\times S_3^2:C_4)$ x 2, $C_2^9.F_9:C_2$ x 2, $C_2^9.F_9:C_2$ x 2, $C_2^{10}.F_9$ x 2, $C_2^8.C_6^2.C_2^3$, $C_2^6.C_6.A_4.C_2^4$, $C_2^8.C_6^2.C_2^3$, $C_2^8.C_6^2.C_2^3$, $C_2^8.C_6^2.C_2^3$

Order 65536: $C_2^6.C_2^5.C_2^4.C_2$

Order 61440: $C_2^{10}.A_5$ x 7, $C_2^{10}.A_5$ x 5, $C_2^9.S_5$ x 2

Order 56320: $C_2^{10}:C_{11}:C_5$

Order 49152: $C_2^8.A_4.C_2^3.C_2$ x 4, $C_2^5.C_2^6.A_4.C_2$ x 2, $C_2^5.C_2^6.S_4$ x 2, $C_2^8.A_4.C_2^3.C_2$ x 2, $C_2^6.C_2^4.C_6.C_2^3$ x 2, $C_2^5.C_2^6.A_4.C_2$, $C_2^8.A_4.C_2^3.C_2$, $C_2^6.C_6.C_2.C_2^6$, $C_2^8.A_4.C_2^4$, $C_2^5.C_2^6.S_4$, $C_2.C_2^6.C_2^6.S_3$, $C_2^8.A_4.C_2^3.C_2$, $C_2^8.A_4.C_2^4$, $C_2^8.A_4.C_2^3.C_2$, $C_2^8.A_4.C_2^3.C_2$, $C_2^8.A_4.C_2^3.C_2$

Order 46080: $C_2^7:A_6$

Order 45056: $C_2^{12}:C_{11}$

Order 40960: $C_2\times C_2^{10}.F_5$ x 6, $C_2\times C_2^8.C_5.C_2^4$

Order 36864: $C_2^6.S_4^2$ x 36, $C_2^6.A_4^2:C_4$ x 14, $C_2^6.S_4^2$ x 13, $C_2^8.C_6^2.C_2^2$ x 12, $C_2^6.C_6.A_4.C_2^3$ x 12, $C_2^9.\SOPlus(4,2)$ x 11, $C_2^8.C_6^2.C_2^2$ x 10, $C_2^9.\SOPlus(4,2)$ x 8, $C_2^8.C_6^2.C_2^2$ x 7, $C_2^6.A_4^2:C_4$ x 6, $C_2^9.\PSU(3,2)$ x 4, $C_2^8.C_3:S_3.D_4$ x 2, $C_2^9.\PSU(3,2)$ x 2, $C_2^9.F_9$ x 2, $C_2^8.F_9:C_2$ x 2, $C_2^6:C_3.A_4.C_2^4$, $C_2^6.C_6^2.C_2^4$, $C_2^6.(C_6\times A_4).C_2^3$, $C_2^8.C_6^2.C_2^2$

Order 32768: $C_2^5.C_2^5.C_2^4.C_2$ x 4, $C_2^5.C_2^5.C_2^4.C_2$ x 3, $C_2^5.C_2^5.C_2^4.C_2$, $C_2^6.C_2^5.C_2^4$, $C_2^5.C_2^5.C_2^4.C_2$, $C_2^{10}.C_4.C_2^3$, $C_2^7.C_2^4.C_2^3.C_2$, $C_2^9.C_2^4.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^5.C_2^4.C_2^4.C_2^2$

Order 30720: $C_2^9.A_5$ x 2, $C_2^8.S_5$

Order 24576: $C_2^6:C_3.D_4^2.C_2$ x 32, $C_2^6.C_6.C_2.C_2^5$ x 14, $C_2^8.A_4.C_2^3$ x 8, $C_2^8.A_4.C_2^3$ x 4, $C_2^8.A_4.D_4$ x 4, $C_2^8.A_4.C_2^3$ x 4, $C_2^8.A_4.C_2^2.C_2$ x 3, $C_2^8.A_4.D_4$ x 3, $C_2^8.A_4.C_2^3$ x 2, $C_2^6.C_6.C_2.C_2^5$ x 2, $C_2^8.A_4.D_4$ x 2, $C_2^6.C_6.C_2.C_2^5$ x 2, $C_2^6.C_6.C_2^6$ x 2, $C_2^6.C_6.C_2^5.C_2$ x 2, $C_2^6.C_6.C_2.C_2^5$ x 2, $C_2^8.A_4.D_4$ x 2, $C_2^8.A_4.D_4$ x 2, $C_2^6.C_6.C_2^6$ x 2, $C_2^5.C_2^3.A_4.C_2^3$ x 2, $C_2^6.C_6.C_2.C_2^5$ x 2, $C_2^8.A_4.C_2^3$, $C_2^8.A_4.C_2^2.C_2$, $C_2^7.A_4.C_2^3.C_2$, $C_2^6.C_6.C_2.C_2^5$, $(C_2^5\times C_4).A_4.C_2.C_2^3$, $C_2^6.C_2^4.C_6.C_2^2$, $C_2^5.A_4.C_2^5.C_2$, $C_2^8.C_6.C_2^4$, $C_2^8.A_4.D_4$, $C_2^6.C_2^4.C_3.D_4$, $C_2^5.A_4.C_2^5.C_2$, $C_2^8.A_4.C_2^2.C_2$, $C_2^8.A_4.C_2^3$, $C_2^5.C_2^6.A_4$, $C_2^6.C_2^4.C_6.C_2^2$, $C_2^5.C_2\wr A_4.C_2^2$, $C_2^4.C_2^4.A_4.C_2^3$, $C_2^6.C_2^4.C_3.D_4$, $C_2^6.C_6.C_2.C_2^5$

Order 23040: $C_2\wr A_6$ x 2, $C_2^6:A_6$

Order 22528: $C_2^{10}.C_{22}$ x 3

Order 20480: $C_2^{10}.D_{10}$ x 18, $C_2^{10}.F_5$ x 12, $C_2^{10}.F_5$ x 4, $C_2^8.C_5.C_2^4$

Order 18432: $C_2^5.S_4^2$ x 38, $C_2^6.\PSOPlus(4,3)$ x 33, $C_2^5.S_4^2$ x 16, $C_2^4.(A_4\times \GL(2,\mathbb{Z}/4))$ x 16, $C_2^6.C_6^2.C_2^3$ x 14, $C_2^5.A_4^2:C_4$ x 14, $C_2^8.\SOPlus(4,2)$ x 10, $C_2^5.S_4^2$ x 10, $C_2^6.(A_4\times S_4)$ x 10, $C_2^6.D_6:D_6.C_2$ x 8, $C_2^8.C_6^2.C_2$ x 5, $C_2^6.(C_6\times A_4).C_2^2$ x 3, $C_2^4.A_4^2.C_2^2.C_2$ x 3, $C_2^6.D_6^2.C_2$ x 2, $C_2^8.F_9$ x 2, $C_2^6.C_6^2.C_2^3$, $A_4^2.C_2^5.C_2^2$, $C_2^6.C_6^2.C_2^3$, $C_2^6.C_6^2.C_2^3$, $C_2^8.C_3^2.Q_8$, $C_2^6.C_6^2.C_2^3$, $C_2^6.C_6^2.C_2^3$, $C_2^4.(C_6\times A_4).C_2^4$, $C_2^6.C_6^2.C_2^3$, $C_2^8.\PSU(3,2)$

Order 16384: $C_2^8.C_2^5.C_2$ x 10, $C_2^8.C_2^5.C_2$ x 4, $C_2^5.C_2^5.C_2^3.C_2$ x 4, $C_2^4.C_2^5.C_2^4.C_2$ x 4, $C_2^8.C_2^5.C_2$ x 4, $C_2^9.C_4.C_2^3$ x 4, $C_2^7.C_2^6.C_2$ x 4, $C_2^5.C_2^4.C_2^4.C_2$ x 4, $C_2^5.C_2^5.C_2^3.C_2$ x 4, $C_2^7.C_2^5.C_2^2$ x 3, $C_2^6.C_2^5.C_2^3$ x 3, $C_2^5.C_2^5.C_2^3.C_2$ x 2, $C_2^4.C_2^5.C_2^4.C_2$ x 2, $C_2^5.C_2^5.C_2^4$ x 2, $C_2^4.C_2^5.C_2^4.C_2$ x 2, $C_2^8.C_2.C_2^5$ x 2, $C_2^7.C_2^3.C_2^3.C_2$ x 2, $C_2^5.C_2^5.C_2^3.C_2$ x 2, $C_2^8.C_2^4.C_2^2$ x 2, $C_2^5.C_2^4.C_2^3.C_2^2$ x 2, $C_2.C_2^6.C_2^4.C_2^3$ x 2, $C_2^7.C_2^6.C_2$, $C_2^8.C_2^5.C_2$, $C_2^8.C_2^5.C_2$, $C_2^8.C_2^4.C_2^2$, $C_2^4.C_2^5.C_2^4.C_2$, $C_2^8.C_2^5.C_2$, $C_2^9.C_4.C_2^3$, $C_2^6.C_2^5.C_2^3$, $C_2^9.C_2^5$, $C_2^7.C_2^6.C_2$, $C_2^4.C_2^5.C_2^3.C_2^2$, $C_2^7.C_2^4.C_2^3$, $C_2^6.C_2^5.C_2^3$, $C_2^7.C_2^5.C_2^2$, $C_2^7.C_2^5.C_2^2$, $C_2^9.C_4.C_2^3$, $C_2^8.C_2^5.C_2$, $C_2^7.C_2^6.C_2$, $C_2^8.C_2.C_2^5$

Order 15840: $C_2\times M_{11}$

Order 15360: $C_2^7.S_5$ x 2, $C_2^7.S_5$ x 2, $C_2^8.A_5$ x 2, $C_2^7.S_5$ x 2, $C_2^7.S_5$, $C_2^7.S_5$

Order 4096: $C_2^{12}$

Order 2048: $C_2^{11}$

Order 11: $C_{11}$

Order 9: $C_3^2$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 2112 (order at least 15360) are shown, as well as all normal subgroups of any index.

Order 32440320: $C_2^{12}.M_{11}$

Order 16220160: $C_2^{11}.M_{11}$

Order 2949120: $C_2^{10}.A_6.D_4$

Order 2703360: $C_2^{12}.\PSL(2,11)$

Order 1474560: $C_2^{12}.A_6$, $C_2^{11}.A_6.C_2$, $C_2^{10}.A_6:C_4$

Order 1351680: $C_2^{11}.\PSL(2,11)$ x 2

Order 737280: $C_2^{11}.A_6$, $C_2^{11}.A_6$, $C_2^{10}.A_6.C_2$

Order 675840: $C_2^{10}.\PSL(2,11)$

Order 589824: $C_2^{12}.F_9:C_2$

Order 491520: $C_2^{12}.S_5$

Order 368640: $C_2^{10}.A_6$

Order 294912: $C_2^{10}.F_9:C_2^2$ x 2, $C_2^9.(C_6^2.\SD_{16})$ x 2, $C_2^8.C_6^2.C_2^4.C_2$, $C_2^{12}.\PSU(3,2)$, $C_2^{12}.F_9$

Order 245760: $C_2^{12}.A_5$ x 2, $C_2^{11}.S_5$ x 2, $C_2^{11}.S_5$ x 2, $C_2^{11}.S_5$, $C_2^{11}.S_5$

Order 225280: $C_2^{12}:C_{11}:C_5$

Order 196608: $C_2^5.C_2^6.A_4.C_2^3$

Order 147456: $C_2^8.C_6^2.C_2^3.C_2$ x 3, $C_2^8.C_3^2.C_2^4.C_2^2$ x 2, $C_2^8.C_6^2.C_2^3.C_2$ x 2, $C_2^{10}.F_9:C_2$ x 2, $C_2^{10}.F_9:C_2$ x 2, $C_2^9.(C_6^2.Q_8)$ x 2, $C_2^{10}.(C_2\times F_9)$ x 2, $C_2^8.C_3^2.C_2^5.C_2$, $C_2^8.(C_2\times D_6:D_6).C_2$, $C_2^8.C_6^2.C_2^4$, $C_2^9.C_6^2:Q_8$

Order 122880: $C_2^{11}.A_5$ x 7, $C_2^{11}.A_5$ x 2, $C_2^{10}.S_5$ x 2, $C_2^{10}.S_5$ x 2, $C_2^{10}.S_5$

Order 112640: $C_2^{10}.C_{11}:C_{10}$ x 2

Order 98304: $C_2^6.C_2.C_2^6.D_6$ x 4, $C_2^6.C_2.C_2^6.D_6$, $C_2^6.C_2^5.A_4.C_2^2$, $C_2^8.A_4.C_2^4.C_2$, $C_2^6.C_2.C_2^6.D_6$

Order 81920: $C_2^{12}.F_5$

Order 73728: $C_2^8.C_6^2.C_2^3$ x 10, $C_2^8.C_3^2.C_2^4.C_2$ x 8, $C_2^{10}.\SOPlus(4,2)$ x 4, $C_2^{10}.\PSU(3,2)$ x 4, $C_2^8.C_3:S_3.C_4^2$ x 3, $C_2^{10}.\SOPlus(4,2)$ x 3, $C_2^8.D_6^2.C_2$ x 2, $C_2^8.(C_2\times S_3^2:C_4)$ x 2, $C_2^9.F_9:C_2$ x 2, $C_2^9.F_9:C_2$ x 2, $C_2^{10}.F_9$ x 2, $C_2^8.C_6^2.C_2^3$, $C_2^6.C_6.A_4.C_2^4$, $C_2^8.C_6^2.C_2^3$, $C_2^8.C_6^2.C_2^3$, $C_2^8.C_6^2.C_2^3$

Order 65536: $C_2^6.C_2^5.C_2^4.C_2$

Order 61440: $C_2^{10}.A_5$ x 5, $C_2^{10}.A_5$ x 4, $C_2^9.S_5$ x 2

Order 56320: $C_2^{10}:C_{11}:C_5$

Order 49152: $C_2^8.A_4.C_2^3.C_2$ x 4, $C_2^5.C_2^6.A_4.C_2$ x 2, $C_2^5.C_2^6.S_4$ x 2, $C_2^8.A_4.C_2^3.C_2$ x 2, $C_2^6.C_2^4.C_6.C_2^3$ x 2, $C_2^5.C_2^6.A_4.C_2$, $C_2^8.A_4.C_2^3.C_2$, $C_2^6.C_6.C_2.C_2^6$, $C_2^8.A_4.C_2^4$, $C_2^5.C_2^6.S_4$, $C_2.C_2^6.C_2^6.S_3$, $C_2^8.A_4.C_2^3.C_2$, $C_2^8.A_4.C_2^4$, $C_2^8.A_4.C_2^3.C_2$, $C_2^8.A_4.C_2^3.C_2$, $C_2^8.A_4.C_2^3.C_2$

Order 46080: $C_2^7:A_6$

Order 45056: $C_2^{12}:C_{11}$

Order 40960: $C_2\times C_2^{10}.F_5$ x 6, $C_2\times C_2^8.C_5.C_2^4$

Order 36864: $C_2^6.S_4^2$ x 20, $C_2^6.S_4^2$ x 10, $C_2^6.A_4^2:C_4$ x 10, $C_2^9.\SOPlus(4,2)$ x 9, $C_2^8.C_6^2.C_2^2$ x 8, $C_2^6.C_6.A_4.C_2^3$ x 8, $C_2^8.C_6^2.C_2^2$ x 8, $C_2^8.C_6^2.C_2^2$ x 6, $C_2^9.\SOPlus(4,2)$ x 6, $C_2^6.A_4^2:C_4$ x 6, $C_2^9.\PSU(3,2)$ x 4, $C_2^9.\PSU(3,2)$ x 2, $C_2^9.F_9$ x 2, $C_2^8.F_9:C_2$ x 2, $C_2^6:C_3.A_4.C_2^4$, $C_2^6.C_6^2.C_2^4$, $C_2^8.C_3:S_3.D_4$, $C_2^6.(C_6\times A_4).C_2^3$, $C_2^8.C_6^2.C_2^2$

Order 32768: $C_2^5.C_2^5.C_2^4.C_2$ x 4, $C_2^5.C_2^5.C_2^4.C_2$ x 3, $C_2^5.C_2^5.C_2^4.C_2$, $C_2^6.C_2^5.C_2^4$, $C_2^5.C_2^5.C_2^4.C_2$, $C_2^{10}.C_4.C_2^3$, $C_2^7.C_2^4.C_2^3.C_2$, $C_2^9.C_2^4.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^5.C_2^4.C_2^4.C_2^2$

Order 30720: $C_2^9.A_5$ x 2, $C_2^8.S_5$

Order 24576: $C_2^6:C_3.D_4^2.C_2$ x 24, $C_2^6.C_6.C_2.C_2^5$ x 7, $C_2^8.A_4.C_2^3$ x 6, $C_2^8.A_4.D_4$ x 4, $C_2^8.A_4.C_2^3$ x 4, $C_2^8.A_4.C_2^2.C_2$ x 3, $C_2^8.A_4.D_4$ x 3, $C_2^8.A_4.C_2^3$ x 2, $C_2^8.A_4.C_2^3$ x 2, $C_2^6.C_6.C_2.C_2^5$ x 2, $C_2^8.A_4.D_4$ x 2, $C_2^6.C_6.C_2.C_2^5$ x 2, $C_2^6.C_6.C_2^6$ x 2, $C_2^6.C_6.C_2.C_2^5$ x 2, $C_2^8.A_4.D_4$ x 2, $C_2^8.A_4.D_4$ x 2, $C_2^6.C_6.C_2^6$ x 2, $C_2^5.C_2^3.A_4.C_2^3$ x 2, $C_2^6.C_6.C_2.C_2^5$ x 2, $C_2^8.A_4.C_2^3$, $C_2^8.A_4.C_2^2.C_2$, $C_2^7.A_4.C_2^3.C_2$, $C_2^6.C_6.C_2^5.C_2$, $C_2^6.C_6.C_2.C_2^5$, $(C_2^5\times C_4).A_4.C_2.C_2^3$, $C_2^6.C_2^4.C_6.C_2^2$, $C_2^5.A_4.C_2^5.C_2$, $C_2^8.C_6.C_2^4$, $C_2^8.A_4.D_4$, $C_2^6.C_2^4.C_3.D_4$, $C_2^5.A_4.C_2^5.C_2$, $C_2^8.A_4.C_2^2.C_2$, $C_2^8.A_4.C_2^3$, $C_2^5.C_2^6.A_4$, $C_2^6.C_2^4.C_6.C_2^2$, $C_2^5.C_2\wr A_4.C_2^2$, $C_2^4.C_2^4.A_4.C_2^3$, $C_2^6.C_2^4.C_3.D_4$, $C_2^6.C_6.C_2.C_2^5$

Order 23040: $C_2^6:A_6$, $C_2\wr A_6$

Order 22528: $C_2^{10}.C_{22}$ x 2

Order 20480: $C_2^{10}.D_{10}$ x 14, $C_2^{10}.F_5$ x 8, $C_2^{10}.F_5$ x 4, $C_2^8.C_5.C_2^4$

Order 18432: $C_2^5.S_4^2$ x 26, $C_2^6.\PSOPlus(4,3)$ x 24, $C_2^5.A_4^2:C_4$ x 14, $C_2^6.C_6^2.C_2^3$ x 10, $C_2^5.S_4^2$ x 10, $C_2^4.(A_4\times \GL(2,\mathbb{Z}/4))$ x 10, $C_2^8.\SOPlus(4,2)$ x 7, $C_2^5.S_4^2$ x 7, $C_2^6.(A_4\times S_4)$ x 7, $C_2^6.D_6:D_6.C_2$ x 5, $C_2^8.C_6^2.C_2$ x 4, $C_2^6.(C_6\times A_4).C_2^2$ x 2, $C_2^4.A_4^2.C_2^2.C_2$ x 2, $C_2^8.F_9$ x 2, $C_2^6.C_6^2.C_2^3$, $A_4^2.C_2^5.C_2^2$, $C_2^6.C_6^2.C_2^3$, $C_2^6.C_6^2.C_2^3$, $C_2^8.C_3^2.Q_8$, $C_2^6.D_6^2.C_2$, $C_2^6.C_6^2.C_2^3$, $C_2^6.C_6^2.C_2^3$, $C_2^4.(C_6\times A_4).C_2^4$, $C_2^6.C_6^2.C_2^3$, $C_2^8.\PSU(3,2)$

Order 16384: $C_2^8.C_2^5.C_2$ x 10, $C_2^8.C_2^5.C_2$ x 4, $C_2^5.C_2^5.C_2^3.C_2$ x 4, $C_2^4.C_2^5.C_2^4.C_2$ x 4, $C_2^8.C_2^5.C_2$ x 4, $C_2^9.C_4.C_2^3$ x 4, $C_2^7.C_2^6.C_2$ x 4, $C_2^5.C_2^4.C_2^4.C_2$ x 4, $C_2^5.C_2^5.C_2^3.C_2$ x 4, $C_2^7.C_2^5.C_2^2$ x 3, $C_2^6.C_2^5.C_2^3$ x 3, $C_2^5.C_2^5.C_2^3.C_2$ x 2, $C_2^4.C_2^5.C_2^4.C_2$ x 2, $C_2^5.C_2^5.C_2^4$ x 2, $C_2^4.C_2^5.C_2^4.C_2$ x 2, $C_2^8.C_2.C_2^5$ x 2, $C_2^7.C_2^3.C_2^3.C_2$ x 2, $C_2^5.C_2^5.C_2^3.C_2$ x 2, $C_2^8.C_2^4.C_2^2$ x 2, $C_2^5.C_2^4.C_2^3.C_2^2$ x 2, $C_2.C_2^6.C_2^4.C_2^3$ x 2, $C_2^7.C_2^6.C_2$, $C_2^8.C_2^5.C_2$, $C_2^8.C_2^5.C_2$, $C_2^8.C_2^4.C_2^2$, $C_2^4.C_2^5.C_2^4.C_2$, $C_2^8.C_2^5.C_2$, $C_2^9.C_4.C_2^3$, $C_2^6.C_2^5.C_2^3$, $C_2^9.C_2^5$, $C_2^7.C_2^6.C_2$, $C_2^4.C_2^5.C_2^3.C_2^2$, $C_2^7.C_2^4.C_2^3$, $C_2^6.C_2^5.C_2^3$, $C_2^7.C_2^5.C_2^2$, $C_2^7.C_2^5.C_2^2$, $C_2^9.C_4.C_2^3$, $C_2^8.C_2^5.C_2$, $C_2^7.C_2^6.C_2$, $C_2^8.C_2.C_2^5$

Order 15840: $C_2\times M_{11}$

Order 15360: $C_2^7.S_5$ x 2, $C_2^7.S_5$ x 2, $C_2^8.A_5$ x 2, $C_2^7.S_5$ x 2, $C_2^7.S_5$, $C_2^7.S_5$

Order 4096: $C_2^{12}$

Order 2048: $C_2^{11}$

Order 11: $C_{11}$

Order 9: $C_3^2$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 32440320: $C_2^{12}.M_{11}$ ($C_1$)

Order 16220160: $C_2^{11}.M_{11}$ ($C_2$)

Order 4096: $C_2^{12}$ ($M_{11}$)

Order 2048: $C_2^{11}$ ($C_2\times M_{11}$)

Order 2: $C_2$ ($C_2^{11}.M_{11}$)

Order 1: $C_1$ ($C_2^{12}.M_{11}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 32440320: $C_2^{12}.M_{11}$ ($C_1$)

Order 16220160: $C_2^{11}.M_{11}$ ($C_2$)

Order 4096: $C_2^{12}$ ($M_{11}$)

Order 2048: $C_2^{11}$ ($C_2\times M_{11}$)

Order 2: $C_2$ ($C_2^{11}.M_{11}$)

Order 1: $C_1$ ($C_2^{12}.M_{11}$)

Series

Derived series	$C_2^{12}.M_{11}$	$\rhd$	$C_2^{11}.M_{11}$
Chief series	$C_2^{12}.M_{11}$	$\rhd$	$C_2^{12}$	$\rhd$	$C_2^{11}$	$\rhd$	$C_2$	$\rhd$	$C_1$
Lower central series	$C_2^{12}.M_{11}$	$\rhd$	$C_2^{11}.M_{11}$
Upper central series	$C_1$	$\lhd$	$C_2$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

The $136 \times 136$ character table is not available for this group.

Rational character table

The $122 \times 122$ rational character table is not available for this group.