Abstract group 56623104.bh: $C_2^{12}.S_4^2:D_{12}$

• Label: 56623104.bh
• Order: \( 2^{21} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{24} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: not computed
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_2^{12}.S_4^2:D_{12}$
Order:	\(56623104\)\(\medspace = 2^{21} \cdot 3^{3} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	Group of order \(452984832\)\(\medspace = 2^{24} \cdot 3^{3} \)
Composition factors:	$C_2$ x 21, $C_3$ x 3
Derived length:	$4$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	295423	90368	12418560	8281856	8257536	24920064	2359296	56623104
Conjugacy classes	1	869	5	1426	161	38	162	2	2664
Divisions	1	869	5	1422	161	36	157	1	2652
Autjugacy classes	1	586	5	627	118	23	67	1	1428

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u \mid d^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,21,17,31,3,23,13,34,6,20,16,35,2,22,18,32,4,24,14,33,5,19,15,36)(7,27,12,26,9,30,8,28,11,25,10,29) \!\cdots\! \rangle$
Transitive group:	36T74487	36T74491			more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^{13}$ . $(S_4^2:D_6)$	$C_2^{17}$ . $(S_3^3:C_2)$	$C_2^{11}$ . $(S_4^3:C_2)$	$C_2^{15}$ . $(D_6^2:D_6)$	all 65

Elements of the group are displayed as permutations of degree 36.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{8}$
Commutator length:	$1$

Subgroups

There are 86 normal subgroups (76 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:	not computed
Frattini:	a subgroup isomorphic to $C_2^3$
Fitting:	not computed
Radical:	not computed
Socle:	not computed

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 18 (order at least 3145728) are shown, as well as all normal subgroups of any index.

Order 56623104: $C_2^{12}.S_4^2:D_{12}$

Order 28311552: $C_2^{16}.C_3.D_6^2$, $C_2^{10}.((C_2\times A_4^2).\GL(2,\mathbb{Z}/4))$, $C_2^{10}.(C_2\times A_4^3:D_4)$, $C_2^{12}.S_4^2:D_6$, $C_2.(C_2^{16}.S_3^3)$, $C_2^{12}.(D_6\times A_4^2):C_4$, $C_2^{12}.S_4^2:C_{12}$

Order 18874368: $C_2^{12}.C_6^2.C_2^6.C_2$

Order 14155776: $C_2^{16}.S_3^3$ x 4, $C_2^{16}.C_3^3.C_2^3$ x 3, $C_2^{12}.A_4^2:D_{12}$ x 2, $C_2^{12}.S_4^2:S_3$ x 2, $C_2^{14}.C_6^2.C_6.C_2^2$, $C_2^{12}.C_6^2.A_4.C_2^3$, $C_2^9.S_4^3:C_2$, $C_2^{10}.A_4^3:D_4$, $C_2^{10}.(C_2\times A_4^3:C_4)$, $C_2^{10}.A_4^3:C_2^3$, $C_2^{10}.A_4^3:D_4$, $C_2^{10}.(C_2\times A_4^3:C_4)$, $C_2^{10}.A_4^3:(C_2\times C_4)$

Order 9437184: $C_2^8.A_4^2.C_2^5.C_2^3$ x 7, $C_2^{12}.C_6^2.C_2^5.C_2$ x 5, $C_2^8.A_4^2.C_2^6.C_2^2$ x 2, $C_2^{14}.C_6^2.C_2^4$, $C_2^{12}.C_6^2.C_2^6$, $C_2^{16}.(S_3\times D_{12})$

Order 7077888: $C_2^{16}.C_3^2.D_6$ x 10, $C_2^{14}.(A_4\times S_3^2)$ x 4, $C_2^{12}.C_6^2.A_4.C_2^2$ x 4, $C_2^{14}.C_3.D_6^2$ x 3, $C_2^{10}.S_4^2:D_6$ x 2, $C_2^{10}.A_4^3:C_2^2$ x 2, $C_2^{14}.C_6^2.C_6.C_2$, $C_2^{12}.C_3.A_4^2.C_2^2$, $C_2^6:C_3.C_2^6.A_4^2.C_2^2$, $C_2^{10}.A_4^3:C_4$, $C_2^8.((C_2\times A_4^2).\GL(2,\mathbb{Z}/4))$, $C_2^9.A_4^3:D_4$, $C_2^9.(A_4^3.C_2^3)$, $C_2^9.(A_4^2:C_4\times S_4)$, $C_2^{10}.S_4^2:C_{12}$, $C_2^{16}.C_3^3.C_4$, $C_2^{10}.A_4^3:C_4$, $C_2^{10}.A_4^3:C_2^2$

Order 6291456: $C_2^{13}.C_6.C_2^6.C_2$

Order 4718592: $C_2^{12}.C_6^2.C_2^5$ x 36, $C_2^8.A_4^2.C_2^6.C_2$ x 24, $C_2^{14}.D_6^2.C_2$ x 20, $C_2^{12}.C_6^2.C_2^4.C_2$ x 18, $C_2^8.A_4^2.C_2^5.C_2^2$ x 12, $C_2^{14}.C_6^2.C_2^3$ x 10, $C_2^8.A_4^2.C_2^4.C_2^3$ x 5, $C_2^{12}.C_6^2.(C_4\times D_4)$ x 4, $C_2^8.A_4^2.C_2.D_4^2$ x 4, $C_2^{12}.C_6^2.C_2^2:D_4$ x 3, $C_2^{12}.C_6^2.C_2.C_2^4$ x 3, $C_2^{12}.A_4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^{12}.C_6.C_2^3.C_6.C_2^2$, $C_2^{12}.C_6.A_4.C_2^4$, $C_2^6.C_6.C_2^6.A_4.C_2^4$, $C_2^6.C_6.C_2^5.A_4.C_2^5$, $C_2^{12}.(C_2\times S_4^2)$, $C_2^{12}.A_4^2:D_4$, $C_2^{12}.A_4:\GL(2,\mathbb{Z}/4)$, $C_2\times C_2^{16}.S_3^2$, $C_2^{12}.(C_4\times A_4\times S_4)$, $C_2^{12}.A_4^2:D_4$

Order 3538944: $C_2^{14}.S_3^3$ x 20, $C_2^{14}.C_3^3.C_2^3$ x 13, $C_2^{12}.C_6^2.C_6.C_2^2$ x 6, $C_2^{10}.S_4^2:S_3$ x 6, $C_2^{16}.C_3^3.C_2$ x 2, $C_2^{14}.C_6^2.C_6$ x 2, $C_2^{12}.C_6^2.A_4.C_2$ x 2, $C_2^6:C_3.C_2^6.A_4^2.C_2$ x 2, $C_2^8.A_4^3:D_4$ x 2, $C_2^9.A_4^3:C_4$ x 2, $C_2^8.A_4^3:C_2^3$ x 2, $C_2^{12}.C_3.A_4^2.C_2$, $C_2^{10}.C_6^2.A_4.C_2^3$, $C_2^8.A_4^2.A_4.C_2^3$, $C_2^8.S_4^2:D_{12}$, $C_2^{10}.D_6^2:D_{12}$, $C_2^8.A_4^3:D_4$, $C_2^8.A_4^3:D_4$, $C_2^9.A_4^3:C_4$, $C_2^9.A_4^3:C_2^2$, $C_2^{10}.A_4^2:S_4$

Order 3145728: $C_2^{13}.C_6.C_2^5.C_2$ x 7, $C_2^{12}.C_6.C_2^5.C_2^2$ x 3, $C_2^{12}.C_6.C_2^3.C_2^3.C_2$ x 2, $C_2^{13}.(C_2^4\times A_4).C_2$, $C_2^{12}.C_6.C_2^6.C_2$, $C_2^{12}.C_6.C_2^4.C_2^3$, $C_2^{12}.C_6.C_2^3.C_2^4$, $C_2^6.C_6.C_2^6.C_2^6.C_2$

Order 2359296: $C_2^8.A_4^2.C_2^6$ x 2, $C_2^{12}.C_6^2.C_2^4$, $C_2^8.A_4^2.C_2^5.C_2$

Order 1769472: $C_2^{16}.C_3^3$

Order 1179648: $C_2^{12}.C_6^2.C_2^3$ x 3, $C_2^8.A_4^2.C_2^5$ x 2, $C_2^{12}.C_3:S_3.C_2^4$

Order 786432: $C_2^{12}.C_6.C_2^5$

Order 589824: $C_2^{12}.D_6^2$ x 2, $C_2^{12}.C_6^2.C_2^2$ x 2

Order 393216: $C_2^{12}.C_6.C_2^4$

Order 294912: $C_2^{12}.C_6^2.C_2$ x 3, $C_2^8.A_4^2.C_2^3$

Order 196608: $C_2^{12}.C_6.C_2^3$

Order 147456: $C_2^{12}.S_3^2$ x 2, $C_2^{12}.C_6^2$, $C_2^{12}.C_3.D_6$

Order 131072: $C_2^{17}$

Order 73728: $C_2^{12}.C_3.S_3$ x 2, $C_2^{12}.C_3.C_6$

Order 65536: $C_2^{16}$

Order 49152: $C_2^{12}.D_6$

Order 36864: $(C_2^6:C_3)^2$

Order 32768: $C_2^{15}$

Order 24576: $C_2^{12}.C_6$

Order 16384: $C_2^{14}$

Order 12288: $C_2^{12}.C_3$

Order 8192: $C_2^{13}$ x 2

Order 4096: $C_2^{12}$ x 2

Order 2048: $C_2^{11}$

Order 1024: $C_2^{10}$

Order 512: $C_2^9$ x 2

Order 256: $C_2^8$ x 2

Order 128: $C_2^7$

Order 64: $C_2^6$

Order 32: $C_2^5$ x 2

Order 16: $C_2^4$ x 2

Order 8: $C_2^3$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 56623104: $C_2^{12}.S_4^2:D_{12}$ ($C_1$)

Order 28311552: $C_2^{16}.C_3.D_6^2$ ($C_2$), $C_2^{10}.((C_2\times A_4^2).\GL(2,\mathbb{Z}/4))$ ($C_2$), $C_2^{10}.(C_2\times A_4^3:D_4)$ ($C_2$), $C_2^{12}.S_4^2:D_6$ ($C_2$), $C_2.(C_2^{16}.S_3^3)$ ($C_2$), $C_2^{12}.(D_6\times A_4^2):C_4$ ($C_2$), $C_2^{12}.S_4^2:C_{12}$ ($C_2$)

Order 14155776: $C_2^{16}.C_3^3.C_2^3$ ($C_2^2$) x 2, $C_2^{12}.C_6^2.A_4.C_2^3$ ($C_2^2$), $C_2^{10}.(C_2\times A_4^3:C_4)$ ($C_2^2$), $C_2^{10}.A_4^3:C_2^3$ ($C_2^2$), $C_2^{10}.(C_2\times A_4^3:C_4)$ ($C_2^2$), $C_2^{10}.A_4^3:(C_2\times C_4)$ ($C_2^2$)

Order 9437184: $C_2^{12}.C_6^2.C_2^5.C_2$ ($S_3$)

Order 7077888: $C_2^{16}.C_3^2.D_6$ ($D_4$) x 3, $C_2^{12}.C_6^2.A_4.C_2^2$ ($D_4$) x 2, $C_2^{12}.C_6^2.A_4.C_2^2$ ($C_2^3$), $C_2^{12}.C_3.A_4^2.C_2^2$ ($D_4$)

Order 4718592: $C_2^8.A_4^2.C_2^6.C_2$ ($D_6$) x 2, $C_2^{12}.C_6^2.C_2^5$ ($D_6$)

Order 3538944: $C_2^{12}.C_6^2.A_4.C_2$ ($C_2\times D_4$) x 2, $C_2^{16}.C_3^3.C_2$ ($C_2\times D_4$)

Order 2359296: $C_2^8.A_4^2.C_2^6$ ($D_{12}$) x 2, $C_2^{12}.C_6^2.C_2^4$ ($C_2\times D_6$), $C_2^8.A_4^2.C_2^5.C_2$ ($S_4$)

Order 1769472: $C_2^{16}.C_3^3$ ($C_2^2\wr C_2$)

Order 1179648: $C_2^{12}.C_6^2.C_2^3$ ($S_3\times D_4$) x 2, $C_2^{12}.C_6^2.C_2^3$ ($C_2\times S_4$), $C_2^{12}.C_3:S_3.C_2^4$ ($C_2\times S_4$), $C_2^8.A_4^2.C_2^5$ ($C_2\times S_4$), $C_2^8.A_4^2.C_2^5$ ($C_2\times D_{12}$)

Order 786432: $C_2^{12}.C_6.C_2^5$ ($\SOPlus(4,2)$)

Order 589824: $C_2^{12}.D_6^2$ ($C_4:S_4$) x 2, $C_2^{12}.C_6^2.C_2^2$ ($C_2^2:D_{12}$), $C_2^{12}.C_6^2.C_2^2$ ($C_2^2\times S_4$)

Order 393216: $C_2^{12}.C_6.C_2^4$ ($S_3^2:C_2^2$)

Order 294912: $C_2^{12}.C_6^2.C_2$ ($D_4\times S_4$) x 2, $C_2^{12}.C_6^2.C_2$ ($C_2^3:D_{12}$), $C_2^8.A_4^2.C_2^3$ ($C_2^3:S_4$)

Order 196608: $C_2^{12}.C_6.C_2^3$ ($D_6\wr C_2$)

Order 147456: $C_2^{12}.S_3^2$ ($C_2^4:D_{12}$) x 2, $C_2^{12}.C_6^2$ ($C_2^4:D_{12}$), $C_2^{12}.C_3.D_6$ ($C_2^4:S_4$)

Order 131072: $C_2^{17}$ ($S_3^3:C_2$)

Order 73728: $C_2^{12}.C_3.S_3$ ($C_2\wr D_6$), $C_2^{12}.C_3.S_3$ ($C_2^5:D_{12}$), $C_2^{12}.C_3.C_6$ ($C_2\wr D_6$)

Order 65536: $C_2^{16}$ ($S_3^2:D_{12}$)

Order 49152: $C_2^{12}.D_6$ ($S_4\wr C_2$)

Order 36864: $(C_2^6:C_3)^2$ ($C_2^6:D_{12}$)

Order 32768: $C_2^{15}$ ($D_6^2:D_6$)

Order 24576: $C_2^{12}.C_6$ ($S_4^2:C_2^2$)

Order 16384: $C_2^{14}$ ($D_6^2:D_{12}$)

Order 12288: $C_2^{12}.C_3$ ($S_4^2:D_4$)

Order 8192: $C_2^{13}$ ($S_4^2:D_6$), $C_2^{13}$ ($D_6^2:(C_2\times S_4)$)

Order 4096: $C_2^{12}$ ($S_4^2:D_{12}$), $C_2^{12}$ ($(C_2^2\times D_6^2):D_{12}$)

Order 2048: $C_2^{11}$ ($S_4^3:C_2$)

Order 1024: $C_2^{10}$ ($(C_2\times S_4^3).C_2$)

Order 512: $C_2^9$ ($A_4^2.C_2\wr D_6$), $C_2^9$ ($(C_2^2:S_4)^2.D_6$)

Order 256: $C_2^8$ ($C_2^3.S_4^3:C_2$), $C_2^8$ ($(C_2^2:S_4)^2.D_{12}$)

Order 128: $C_2^7$ ($C_2^8.D_6^2:D_6$)

Order 64: $C_2^6$ ($C_2^8.D_6^2:D_{12}$)

Order 32: $C_2^5$ ($C_2^8.S_4^2:D_6$), $C_2^5$ ($C_2^{10}.D_6^2:D_6$)

Order 16: $C_2^4$ ($C_2^8.S_4^2:D_{12}$), $C_2^4$ ($C_2^{10}.D_6^2:D_{12}$)

Order 8: $C_2^3$ ($C_2^{10}.S_4^2:D_6$)

Order 4: $C_2^2$ ($C_2^9.S_4^3:C_2$)

Order 2: $C_2$ ($C_2^{16}.S_3^3.C_2$)

Order 1: $C_1$ ($C_2^{12}.S_4^2:D_{12}$)

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 4 larger groups in the database.

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

Character theory

Complex character table

The $2664 \times 2664$ character table is not available for this group.

Rational character table

The $2652 \times 2652$ rational character table is not available for this group.