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

• Label: 32440320.f
• Order: \( 2^{16} \cdot 3^{2} \cdot 5 \cdot 11 \)
• Exponent: \( 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3^{3} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $24$
• Trans deg.: $44$
• 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:	$C_2^{11}.M_{11}\times S_3$, of order \(97320960\)\(\medspace = 2^{16} \cdot 3^{3} \cdot 5 \cdot 11 \)
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	35	1	48	1	27	44	15	2	12	16	6	208
Divisions	1	35	1	48	1	27	32	15	1	12	8	3	184
Autjugacy classes	1	17	1	22	1	13	12	7	2	6	4	2	88

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$44$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $24$ $\langle(1,3)(2,5)(4,8,14,9,16,22,15,17)(6,10,13,7,12,20,11,18)(19,21)(23,24), (1,2,4,7,11,17,5,3,6,9,15,18)(8,13)(10,14)(12,19,20)(16,21,22)\rangle$
Transitive group:	44T815	44T819			more information
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(C_2^{10}.M_{11})$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^{12}$ . $M_{11}$	$(C_2^{11}.M_{11})$ . $C_2$	$C_2^{11}$ . $(C_2\times M_{11})$	$C_2$ . $(C_2^{11}.M_{11})$	all 7

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

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 15 normal subgroups (6 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^{10}.M_{11}$
Commutator:	$G' \simeq$ $C_2^{10}.M_{11}$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^{12}.M_{11}$
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:	$\operatorname{soc} \simeq$ $C_2^{12}$	$G/\operatorname{soc} \simeq$ $M_{11}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5.C_2^3.C_2^6.C_2^2$
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 396 (order at least 81920) are shown, as well as all normal subgroups of any index.

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

Order 16220160: $C_2^{11}.M_{11}$ x 3

Order 8110080: $C_2^{10}.M_{11}$

Order 2949120: $C_2\times C_2^{11}.A_6.C_2$

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

Order 1474560: $C_2^{11}.A_6.C_2$ x 12, $C_2^{11}.A_6.C_2$ x 2, $C_2^{12}.A_6$

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

Order 737280: $C_2^{10}.A_6.C_2$ x 16, $C_2^{10}.A_6.C_2$ x 12, $C_2^{11}.A_6$ x 6, $C_2^{11}.A_6$

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

Order 589824: $C_2^8.C_3:S_3.C_2^5.C_2^2$

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

Order 368640: $C_2^9.A_6.C_2$ x 8, $C_2^{10}.A_6$ x 4, $C_2^{10}.A_6$ x 3

Order 294912: $C_2^8.C_3:S_3.C_2^3.C_2^3$ x 15, $C_2^8.C_3:S_3.C_2^4.C_2^2$ x 6, $C_2^8.C_3:S_3.C_2^5.C_2$ x 3, $C_2^8.C_3^2.C_4.C_2^5$ x 2, $C_2^8.C_3:S_3.C_2^3.C_2^3$ x 2, $C_2\times C_2^8.C_3:S_3.C_2^5$, $C_2^8.C_3^2.C_2.C_4.C_2^4$, $C_2^8.C_3^2.C_2^4.C_2^3$

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

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

Order 196608: $C_2^4.C_2^3:Q_8.A_4.C_2^4$

Order 184320: $C_2^9.A_6$

Order 147456: $C_2^{10}.F_9:C_2$ x 64, $C_2^{10}.F_9:C_2$ x 40, $C_2\times C_2^8.C_3:S_3.C_2^4$ x 34, $C_2^8.C_3^2.Q_8:C_4.C_2$ x 18, $C_2^8.D_6:D_6.C_2^2$ x 12, $C_2^8.C_2^2:F_9.C_2$ x 6, $C_2^8.(C_2^2\times F_9).C_2$ x 6, $C_2^8.C_3:S_3.C_2^4.C_2$ x 6, $C_2^8.D_6^2.C_2^2$ x 3, $C_2^8.(C_2\times D_6:D_6).C_2$ x 3, $C_2^2\times C_2^8.C_3:S_3.C_2^3$ x 3, $C_2^2\times C_2^8.C_3:S_3.C_2^3$ x 2, $C_2\times C_2^8.(C_2^2\times F_9)$ x 2, $C_2\times C_2^8.C_3:S_3.C_2^4$, $C_2^8.C_3:S_3.C_2.C_2^4$, $C_2^8.C_6^2.C_2^4$, $C_2^8.C_3:S_3.C_2.C_2^4$

Order 122880: $C_2^{10}.S_5$ x 16, $C_2^{11}.A_5$ x 12, $C_2^{10}.S_5$ x 6, $C_2^{10}.S_5$ x 6, $C_2^{11}.A_5$ x 2

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

Order 98304: $C_2^5.C_2^5.A_4.C_2^3$ x 12, $C_2^5.C_2^5.A_4.C_2^3$, $C_2^5.C_2^5.A_4.C_2^3$, $C_2^8.A_4.C_2^5$, $C_2^5.C_2^5.A_4.C_2^3$

Order 81920: $C_2^8.D_5.C_2^5$

Order 65536: $C_2^5.C_2^3.C_2^6.C_2^2$

Order 31680: $C_2^2\times M_{11}$ x 2

Order 4096: $C_2^{12}$

Order 2048: $C_2^{11}$ x 3

Order 1024: $C_2^{10}$

Order 11: $C_{11}$

Order 9: $C_3^2$

Order 5: $C_5$

Order 4: $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 396 (order at least 81920) 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 8110080: $C_2^{10}.M_{11}$

Order 2949120: $C_2\times C_2^{11}.A_6.C_2$

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

Order 1474560: $C_2^{11}.A_6.C_2$ x 3, $C_2^{11}.A_6.C_2$, $C_2^{12}.A_6$

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

Order 737280: $C_2^{10}.A_6.C_2$ x 4, $C_2^{10}.A_6.C_2$ x 2, $C_2^{11}.A_6$ x 2, $C_2^{11}.A_6$

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

Order 589824: $C_2^8.C_3:S_3.C_2^5.C_2^2$

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

Order 368640: $C_2^9.A_6.C_2$ x 2, $C_2^{10}.A_6$ x 2, $C_2^{10}.A_6$

Order 294912: $C_2^8.C_3:S_3.C_2^3.C_2^3$ x 4, $C_2^8.C_3:S_3.C_2^5.C_2$, $C_2^8.C_3:S_3.C_2^4.C_2^2$, $C_2^8.C_3^2.C_4.C_2^5$, $C_2\times C_2^8.C_3:S_3.C_2^5$, $C_2^8.C_3^2.C_2.C_4.C_2^4$, $C_2^8.C_3:S_3.C_2^3.C_2^3$, $C_2^8.C_3^2.C_2^4.C_2^3$

Order 245760: $C_2^{11}.S_5$ x 4, $C_2^{12}.A_5$, $C_2\times C_2^{11}.A_5$, $C_2^{11}.S_5$, $C_2^{11}.S_5$

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

Order 196608: $C_2^4.C_2^3:Q_8.A_4.C_2^4$

Order 184320: $C_2^9.A_6$

Order 147456: $C_2^{10}.F_9:C_2$ x 11, $C_2\times C_2^8.C_3:S_3.C_2^4$ x 10, $C_2^{10}.F_9:C_2$ x 8, $C_2^8.D_6:D_6.C_2^2$ x 4, $C_2^8.C_3^2.Q_8:C_4.C_2$ x 3, $C_2^2\times C_2^8.C_3:S_3.C_2^3$ x 2, $C_2^8.C_2^2:F_9.C_2$ x 2, $C_2^8.(C_2^2\times F_9).C_2$, $C_2\times C_2^8.(C_2^2\times F_9)$, $C_2\times C_2^8.C_3:S_3.C_2^4$, $C_2^8.C_3:S_3.C_2.C_2^4$, $C_2^8.C_3:S_3.C_2^4.C_2$, $C_2^8.C_6^2.C_2^4$, $C_2^8.D_6^2.C_2^2$, $C_2^8.(C_2\times D_6:D_6).C_2$, $C_2^2\times C_2^8.C_3:S_3.C_2^3$, $C_2^8.C_3:S_3.C_2.C_2^4$

Order 122880: $C_2^{10}.S_5$ x 5, $C_2^{11}.A_5$ x 4, $C_2^{11}.A_5$ x 2, $C_2^{10}.S_5$ x 2, $C_2^{10}.S_5$ x 2

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

Order 98304: $C_2^5.C_2^5.A_4.C_2^3$ x 4, $C_2^5.C_2^5.A_4.C_2^3$, $C_2^5.C_2^5.A_4.C_2^3$, $C_2^8.A_4.C_2^5$, $C_2^5.C_2^5.A_4.C_2^3$

Order 81920: $C_2^8.D_5.C_2^5$

Order 65536: $C_2^5.C_2^3.C_2^6.C_2^2$

Order 31680: $C_2^2\times M_{11}$

Order 4096: $C_2^{12}$

Order 2048: $C_2^{11}$

Order 1024: $C_2^{10}$

Order 11: $C_{11}$

Order 9: $C_3^2$

Order 5: $C_5$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 16220160: $C_2^{11}.M_{11}$ ($C_2$) x 3

Order 8110080: $C_2^{10}.M_{11}$ ($C_2^2$)

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

Order 2048: $C_2^{11}$ ($C_2\times M_{11}$) x 3

Order 1024: $C_2^{10}$ ($C_2^2\times M_{11}$)

Order 4: $C_2^2$ ($C_2^{10}.M_{11}$)

Order 2: $C_2$ ($C_2^{11}.M_{11}$) x 3

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 8110080: $C_2^{10}.M_{11}$ ($C_2^2$)

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

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

Order 1024: $C_2^{10}$ ($C_2^2\times M_{11}$)

Order 4: $C_2^2$ ($C_2^{10}.M_{11}$)

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

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

Series

Derived series	$C_2^{12}.M_{11}$	$\rhd$	$C_2^{10}.M_{11}$
Chief series	$C_2^{12}.M_{11}$	$\rhd$	$C_2^{12}$	$\rhd$	$C_2^{11}$	$\rhd$	$C_2^{10}$	$\rhd$	$C_1$
Lower central series	$C_2^{12}.M_{11}$	$\rhd$	$C_2^{10}.M_{11}$
Upper central series	$C_1$	$\lhd$	$C_2^2$

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

The $208 \times 208$ character table is not available for this group.

Rational character table

The $184 \times 184$ rational character table is not available for this group.