Abstract group 33218887680.a: $C_2^{22}.M_{11}$

• Label: 33218887680.a
• Order: \( 2^{26} \cdot 3^{2} \cdot 5 \cdot 11 \)
• Exponent: \( 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{27} \cdot 3^{2} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $44$
• Trans deg.: $44$
• Rank: $2$

Group information

Description:	$C_2^{22}.M_{11}$
Order:	\(33218887680\)\(\medspace = 2^{26} \cdot 3^{2} \cdot 5 \cdot 11 \)
Exponent:	\(2640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	Group of order \(66437775360\)\(\medspace = 2^{27} \cdot 3^{2} \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 22, $M_{11}$
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	6897663	1802240	948879360	103809024	2189721600	5969018880	6539968512	1509949440	5190451200	6228541440	4529848320	33218887680
Conjugacy classes	1	998	1	995	1	227	134	63	2	60	24	6	2512
Divisions	1	998	1	994	1	203	103	39	1	60	12	3	2416
Autjugacy classes	1	446	1	431	1	104	68	29	2	27	16	4	1130

Minimal presentations

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

Minimal degrees of linear representations for this group have not been computed

Constructions

Permutation group:	Degree $44$ $\langle(1,5,20,24,2,6,19,23)(3,7,18,22)(4,8,17,21)(9,31,36,27)(10,32,35,28)(11,29,33,26) \!\cdots\! \rangle$
Transitive group:	44T1731	44T1742			more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$C_2^{22}$ . $M_{11}$	$(C_2^{21}.M_{11})$ . $C_2$	$C_2^{21}$ . $(C_2\times M_{11})$	$C_2$ . $(C_2^{21}.M_{11})$	all 8

Elements of the group are displayed as permutations of degree 44.

Homology

Abelianization:	$C_{2} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	not computed

Subgroups

There are 11 normal subgroups (9 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_2$
Commutator:	a subgroup isomorphic to $C_2^{21}.M_{11}$
Frattini:	not computed
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2^6.C_2^5.C_2^6$
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 792 (order at least 41943040) are shown, as well as all normal subgroups of any index.

Order 33218887680: $C_2^{22}.M_{11}$

Order 16609443840: $C_2^{21}.M_{11}$

Order 3019898880: $C_2^{22}.A_6.C_2$

Order 2768240640: $C_2^{22}.\PSL(2,11)$

Order 1509949440: $C_2^{21}.A_6.C_2$ x 6, $C_2^{22}.A_6$

Order 1384120320: $C_2^{21}.\PSL(2,11)$ x 3

Order 754974720: $C_2^{21}.A_6$ x 5, $C_2^{20}.A_6.C_2$ x 4, $C_2^{20}.A_6.C_2$ x 2

Order 692060160: $C_2^{20}.\PSL(2,11)$

Order 603979776: $C_2^{16}.C_3^2.C_2^5.C_2^4.C_2$

Order 503316480: $C_2^{22}.S_5$

Order 377487360: $C_2^{20}.A_6$ x 3, $C_2^{20}.A_6$ x 2, $C_2^{19}.A_6.C_2$, $C_2^{19}.A_6.C_2$

Order 301989888: $C_2^{16}.C_3^2.C_2^5.C_2^3.C_2$ x 4, $C_2^{16}.C_3^2.C_2^4.C_2^4.C_2$ x 4, $C_2^{16}.C_6^2.C_2^5.C_2^2$ x 2, $C_2^{16}.C_3^2.C_2^4.C_2^3.C_2^2$ x 2, $C_2^{16}.C_3:S_3.C_2^5.C_2^3$ x 2, $C_2^{16}.C_6^2.C_2^3.C_2^3.C_2$

Order 251658240: $C_2^{21}.S_5$ x 14, $C_2^{22}.A_5$ x 2

Order 230686720: $C_2^{20}.C_{11}.C_{10}.C_2$

Order 201326592: $C_2^{13}.C_2.C_2^6.A_4.C_2^4$

Order 188743680: $C_2^{19}.A_6$

Order 150994944: $C_2^{16}.C_6^2.C_2^4.C_2^2$ x 23, $C_2^{16}.C_3:S_3.C_2^4.C_2^3$ x 12, $C_2^{16}.C_6^2.C_2^5.C_2$ x 8, $C_2^{16}.C_3^2.C_2^4.C_2^3.C_2$ x 8, $C_2^{16}.C_6^2.C_2.C_4.C_2^3$ x 6, $C_2^{16}.C_3:S_3.C_2^5.C_2^2$ x 4, $C_2^{16}.C_3:S_3.C_2^3.C_4^2$ x 4, $C_2^{16}.C_3:S_3.C_2.C_2^4.C_2^2$ x 4, $C_2^{16}.C_3^2.C_2^6.C_2^2$ x 3, $C_2^{16}.C_6^2.C_2.C_2^5$ x 2, $C_2^{16}.C_6^2.C_2^6$

Order 125829120: $C_2^{20}.S_5$ x 32, $C_2^{21}.A_5$ x 26

Order 115343360: $C_2^{20}.C_{11}.C_{10}$ x 3

Order 100663296: $C_2^{13}.C_2.C_2^6.A_4.C_2^3$ x 14, $C_2^{16}.A_4.C_2^6.C_2$, $C_2^9.C_2^5.C_2^6.A_4.C_2^3$

Order 83886080: $C_2^{16}.C_{10}.C_2^6.C_2$

Order 75497472: $C_2^{16}.C_3^2.C_2^5.C_2^2$ x 73, $C_2^{16}.C_6^2.C_2^4.C_2$ x 38, $C_2^{16}.C_6^2.C_2^5$ x 37, $C_2^{16}.C_6^2.C_2^3.C_2^2$ x 26, $C_2^{16}.C_6^2.C_2.C_2^4$ x 24, $C_2^{16}.C_3:S_3.C_2^4.C_2^2$ x 22, $C_2^{16}.C_6^2.C_2^2:D_4$ x 20, $C_2^{16}.C_3^2.C_2^6.C_2$ x 11, $C_2^{16}.C_3:S_3.C_2^3.C_2^3$ x 10, $C_2^{16}.C_3:S_3.C_2\wr C_2^2$ x 10, $C_2^{16}.C_6^2.C_4.C_2^3$ x 8, $C_2^{16}.C_6^2.C_2^2\wr C_2$ x 6, $C_2^{16}.C_6^2.C_2^2:C_8$ x 6, $C_2^{16}.C_3:S_3.C_2^3.C_4.C_2$ x 4, $C_2^{16}.C_3:S_3.C_2\wr C_4$ x 4, $C_2^{16}.C_3:S_3.C_2^5.C_2$ x 2, $C_2^{12}.C_3.C_2^6.C_6.C_2^4$, $C_2\times C_2^{16}.C_6^2.C_2^4$

Order 67108864: $C_2^9.C_2^6.C_2^5.C_2^6$

Order 62914560: $C_2^{20}.A_5$ x 52, $C_2^{19}.S_5$ x 20, $C_2^{20}.A_5$ x 6

Order 57671680: $C_2^{20}.C_{11}.C_5$

Order 50331648: $C_2^{16}.A_4.C_2^5.C_2$ x 59, $C_2^{13}.C_2^5.C_2^4.D_6$ x 28, $C_2^{13}.C_2^5.C_2^4.C_6.C_2$ x 7, $C_2^{16}.A_4.C_2^6$ x 2, $C_2^{14}.C_2^4.C_6.C_2^5$ x 2, $C_2^{10}.C_2.C_2^6.A_4.C_2^5$ x 2, $C_2^{12}.C_6.C_2^6.C_2^5$

Order 47185920: $C_2^{17}.A_6$

Order 46137344: $C_2^{20}.C_{22}.C_2$

Order 41943040: $C_2^{16}.C_5.C_2^6.C_2$ x 24, $C_2^{16}.D_{10}.C_2^5$ x 6, $C_2^{16}.C_{10}.C_2^6$

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

Order 4194304: $C_2^{22}$

Order 2097152: $C_2^{21}$

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 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 33218887680: $C_2^{22}.M_{11}$ ($C_1$)

Order 16609443840: $C_2^{21}.M_{11}$ ($C_2$)

Order 4194304: $C_2^{22}$ ($M_{11}$)

Order 2097152: $C_2^{21}$ ($C_2\times M_{11}$)

Order 4096: $C_2^{12}$ ($C_2^{10}.M_{11}$)

Order 2048: $C_2^{11}$ ($C_2^{11}.M_{11}$) x 2, $C_2^{11}$ ($C_2^{11}.M_{11}$)

Order 1024: $C_2^{10}$ ($C_2^{12}.M_{11}$)

Order 2: $C_2$ ($C_2^{21}.M_{11}$)

Order 1: $C_1$ ($C_2^{22}.M_{11}$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

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

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $2512 \times 2512$ character table is not available for this group.

Rational character table

The $2416 \times 2416$ rational character table is not available for this group.