Abstract group 16609443840.b: $C_2^{20}.C_2.M_{11}$

• Label: 16609443840.b
• Order: \( 2^{25} \cdot 3^{2} \cdot 5 \cdot 11 \)
• Exponent: \( 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{26} \cdot 3^{2} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $44$
• Trans deg.: $44$
• Rank: $2$

Group information

Description:	$C_2^{20}.C_2.M_{11}$
Order:	\(16609443840\)\(\medspace = 2^{25} \cdot 3^{2} \cdot 5 \cdot 11 \)
Exponent:	\(2640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	Group of order \(33218887680\)\(\medspace = 2^{26} \cdot 3^{2} \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 21, $M_{11}$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	10	11	12	16	20	22
Elements	1	1894399	1802240	475994112	103809024	661422080	2984509440	1972371456	1509949440	3027763200	3114270720	1245708288	1509949440	16609443840
Conjugacy classes	1	161	1	210	1	36	100	10	2	23	28	3	2	578
Divisions	1	161	1	210	1	33	73	10	1	22	13	3	1	530
Autjugacy classes	1	161	1	208	1	36	66	10	2	23	16	3	2	530

Minimal presentations

Permutation degree:	$44$
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	22	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $44$ $\langle(1,10,6,36,29,13,2,9,5,35,30,14)(3,11,8,34,31,15,4,12,7,33,32,16)(17,24,26,19,22,28,18,23,25,20,21,27) \!\cdots\! \rangle$
Transitive group:	44T1707	44T1708			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^{20}.M_{11})$ . $C_2$	$(C_2^{20}.C_2)$ . $M_{11}$	$C_2^{20}$ . $(C_2\times M_{11})$	$C_2^{11}$ . $(C_2^{10}.M_{11})$	all 5

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

Homology

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

Subgroups

There are 7 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $C_2^{20}.M_{11}$
Frattini:	a subgroup isomorphic to $C_2^{10}$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2^6.C_2^5.C_2^5$
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 20971520) are shown, as well as all normal subgroups of any index.

Order 16609443840: $C_2^{20}.C_2.M_{11}$

Order 8304721920: $C_2^{20}.M_{11}$

Order 1509949440: $C_2^{19}.A_6.C_2^3$

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

Order 754974720: $C_2^{18}.A_6.D_4$, $C_2^{18}.A_6.D_4$, $C_2^{20}.C_2.A_6$, $C_2^{20}.A_6.C_2$, $C_2^{18}.A_6.D_4$, $C_2^{18}.A_6.C_2^3$, $C_2^{19}.A_6.C_2^2$

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

Order 377487360: $C_2^{19}.A_6.C_2$ x 4, $C_2^{18}.A_6.C_2^2$ x 2, $C_2^{19}.C_2.A_6$, $C_2^{18}.A_6.C_4$, $C_2^{18}.A_6.C_4$, $C_2^{19}.A_6.C_2$, $C_2^{20}.A_6$

Order 301989888: $C_2^{16}.C_3^2.C_2^3.D_4^2$

Order 251658240: $C_2^{20}.C_2.S_5$

Order 188743680: $C_2^{18}.A_6.C_2$ x 3, $C_2^{19}.A_6$ x 2, $C_2^{18}.C_2.A_6$, $C_2^{18}.A_6.C_2$

Order 150994944: $C_2^{16}.C_3:S_3.C_2^4.C_2^3$ x 6, $C_2^{16}.C_3:S_3.C_2^5.C_2^2$ x 3, $C_2^{16}.C_3^2.C_2^4.D_4.C_2$ x 2, $C_2^{16}.C_6^2.D_4^2$, $C_2^{16}.C_6^2.C_2^2:C_8.C_2$, $C_2^{16}.C_6^2.C_2.C_2^5$, $C_2^{16}.C_3:S_3.C_2^6.C_2$

Order 125829120: $C_2^{19}.C_2.S_5$ x 3, $C_2^{20}.C_2.A_5$ x 2, $C_2^{20}.S_5$, $C_2^{15}.C_2^5.S_5$, $C_2^{20}.S_5$

Order 115343360: $(C_2^{20}.C_{22}):C_5$

Order 100663296: $C_2^9.C_2^6.C_2^6.C_6.C_2^3$

Order 94371840: $C_2^{18}.A_6$

Order 75497472: $C_2^{16}.C_3:S_3.C_2^4.C_2^2$ x 16, $C_2^{16}.C_6^2.C_2.C_2^4$ x 15, $C_2^{16}.C_3:S_3.C_2^5.C_2$ x 8, $C_2^{16}.C_3:S_3.C_2^3.C_2^3$ x 8, $C_2^{16}.C_3^2.C_2^4.C_2^3$ x 5, $C_2^{16}.C_3:S_3.C_2\wr C_2^2$ x 4, $C_2^{16}.C_6^2.C_4.C_2^3$ x 3, $C_2^{16}.C_6^2.C_2.C_4^2$ x 2, $C_2^{16}.C_6^2.(C_4\times D_4)$ x 2, $C_2^{16}.C_3:S_3.D_4^2$ x 2, $C_2^{17}.C_6^2.\SD_{16}$ x 2, $C_2^{16}.C_6^2.D_4.C_2^2$, $C_2^{16}.C_6^2.C_4:C_4.C_2$, $C_2^{16}.C_6^2.C_2^4.C_2$, $C_2^{16}.C_6^2.C_2^2\wr C_2$, $C_2^{16}.C_6^2.C_2^2:D_4$, $C_2^{16}.C_6^2.C_2^2.C_2^3$, $C_2^{16}.C_3^2.C_2.C_4.C_2^4$, $C_2^{16}.C_3:S_3.C_2^6$

Order 62914560: $C_2^{19}.S_5$ x 4, $C_2^{18}.A_5.C_4$ x 2, $C_2^{19}.A_5.C_2$ x 2, $C_2^{20}.A_5$, $C_2^{18}.C_2.S_5$, $C_2^{20}.A_5$, $C_2^{18}.C_2.S_5$, $C_2^{19}.S_5$, $C_2^{19}.S_5$

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

Order 50331648: $C_2^9.C_2^6.C_2^6.D_6.C_2$ x 3, $C_2^{16}.A_4.C_2^5.C_2$, $C_2^{12}.C_2^6.C_2^4.C_6.C_2$, $C_2^9.C_2^6.C_2^6.D_{12}$, $C_2^9.C_2^6.C_2^6.C_6.C_2^2$, $C_2^9.C_2^6.C_2^6.C_3.D_4$

Order 41943040: $C_2^{12}.(C_2^9.F_5)$

Order 37748736: $C_2^{16}.C_3:S_3.C_2^4.C_2$ x 56, $C_2^{16}.C_3:S_3.C_2^5$ x 24, $C_2^{16}.C_3^2.C_2^4.C_2^2$ x 18, $C_2^{16}.C_3^2.C_2^3.C_2^3$ x 16, $C_2^{16}.C_6^2.\SD_{16}$ x 16, $C_2^{16}.C_6^2.C_2.C_2^3$ x 15, $C_2^{16}.C_3^2.C_4.D_4.C_2$ x 14, $C_2^{17}.C_6^2.Q_8$ x 14, $C_2^{16}.C_3:S_3.(C_4\times D_4)$ x 10, $C_2^{16}.C_3^2.C_2.C_4.C_2^3$ x 8, $C_2^{16}.C_3:S_3.C_2.C_2^4$ x 8, $C_2^{16}.C_3^2.C_2.C_4:Q_8$ x 5, $C_2^{16}.C_2^2:F_9.C_2$ x 5, $C_2^{16}.C_6^2.C_2^4$ x 4, $C_2^{17}.C_6^2.C_8$ x 4, $C_2^{16}.C_6^2.C_2^3.C_2$ x 3, $C_2^{16}.C_3^2.C_4.C_2^4$ x 3, $C_2^{16}.C_2^2:S_3\wr C_2.C_2$ x 3, $C_2^{16}.F_9.C_2^3$ x 3, $C_2^{16}.C_6^2.D_4.C_2$ x 2, $C_2^{16}.C_6^2.C_8.C_2$ x 2, $C_2^{16}.C_3^2.Q_8:Q_8$ x 2, $C_2^{16}.C_3^2.Q_8:C_4.C_2$ x 2, $C_2^{16}.C_3^2.C_4.D_4:C_2$ x 2, $C_2^{16}.(C_2\times D_6:D_6).C_2$ x 2, $C_2^{16}.C_6^2.\SD_{16}$ x 2, $C_2^{16}.(C_4\times F_9).C_2$ x 2, $C_2^{16}.C_6^2.\SD_{16}$ x 2, $C_2^{16}.F_9.D_4$ x 2, $C_2^{16}.F_9.D_4$ x 2, $C_2^{12}.(C_4\times A_4^2).\SD_{16}$ x 2, $C_2^{16}.F_9.C_2^3$ x 2, $C_2^{12}.S_4\wr C_2.D_4$ x 2, $C_2^{16}.F_9.D_4$ x 2, $C_2^{16}.C_3^2.D_4^2$, $C_2^{16}.C_3^2.C_2^2:Q_8.C_2$, $C_2^{16}.C_3^2.C_2^2.C_2^4$, $C_2^{16}.C_3:S_3.C_2^2\wr C_2$, $C_2^{16}.C_3:S_3.C_2^2:D_4$, $C_2^{12}.C_6.C_2^6.C_6.C_2^2$, $C_2^{16}.C_6^2.\SD_{16}$, $C_2^{12}.A_4^2:D_4^2$

Order 33554432: $C_2^9.C_2^6.C_2^5.C_2^5$

Order 31457280: $C_2^{19}.A_5$ x 3, $C_2^{18}.S_5$ x 2, $C_2^{18}.C_2.A_5$ x 2, $C_2^{19}.A_5$, $C_2^{18}.S_5$, $C_2^{18}.S_5$

Order 25165824: $C_2^{16}.A_4.C_2^4.C_2$ x 11, $C_2^9.C_2^6.C_2^6.D_6$ x 7, $C_2^9.C_2^6.C_2^6.C_6.C_2$ x 3, $C_2^{10}.C_2.C_2^6.(D_4\times S_4)$ x 2, $C_2^{18}.A_4.C_2^3$, $C_2^{16}.A_4.C_2^5$, $C_2^{14}.C_2^4.C_2^4:S_3$, $C_2^{12}.C_6.C_2^6.C_2^4$, $C_2^{12}.C_2^6.C_6.C_2^4$, $C_2^9.C_2^6.C_2^6.C_{12}$

Order 23592960: $C_2^{10}.C_2^6:A_6$

Order 23068672: $C_2^{20}.C_{22}$

Order 20971520: $C_2^{16}.D_{10}.C_4^2$ x 3, $C_2^{10}.(C_2^{10}.F_5)$ x 3, $C_2^{10}.(C_2^{10}.F_5)$ x 3, $C_2^{16}.D_{10}.C_2^4$ x 3, $C_2\times C_2^{16}.D_5.C_2^4$, $C_2^4.(C_2^{16}.D_{10})$, $C_2^{16}.C_5.C_2^5.C_2$

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

Order 2097152: $C_2^{20}.C_2$

Order 1048576: $C_2^{20}$

Order 2048: $C_2^{11}$

Order 1024: $C_2^{10}$

Order 11: $C_{11}$

Order 9: $C_3^2$

Order 5: $C_5$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 16609443840: $C_2^{20}.C_2.M_{11}$ ($C_1$)

Order 8304721920: $C_2^{20}.M_{11}$ ($C_2$)

Order 2097152: $C_2^{20}.C_2$ ($M_{11}$)

Order 1048576: $C_2^{20}$ ($C_2\times M_{11}$)

Order 2048: $C_2^{11}$ ($C_2^{10}.M_{11}$)

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

Order 1: $C_1$ ($C_2^{20}.C_2.M_{11}$)

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 3 larger groups in the database.

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

Character theory

Complex character table

The $578 \times 578$ character table is not available for this group.

Rational character table

The $530 \times 530$ rational character table is not available for this group.