Abstract group 1632586752.bbj: $C_3^6.(C_3^6.C_2^7:S_4)$

• Label: 1632586752.bbj
• Order: \( 2^{10} \cdot 3^{13} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{13} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: not computed
• Trans deg.: $36$
• Rank: $4$

Group information

Description:	$C_3^6.(C_3^6.C_2^7:S_4)$
Order:	\(1632586752\)\(\medspace = 2^{10} \cdot 3^{13} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	Group of order \(6530347008\)\(\medspace = 2^{12} \cdot 3^{13} \)
Composition factors:	$C_2$ x 10, $C_3$ x 13
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	9	12	18
Elements	1	1375839	1371248	56477088	352701648	67184640	810834624	342641664	1632586752
Conjugacy classes	1	37	487	44	2093	19	462	27	3170
Divisions	1	37	487	44	2093	19	460	27	3168
Autjugacy classes	1	27	233	28	1043	9	232	12	1585

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$36$
Rank:	$4$
Inequivalent generating quadruples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k, l, m, n, o \mid d^{6}=e^{6}=f^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,13,22)(2,15,24)(3,14,23)(4,16,33,7,20,30)(5,17,32,8,21,29)(6,18,31,9,19,28) \!\cdots\! \rangle$
Transitive group:	36T96782	36T96922			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:	$C_3^{12}$ . $(C_2^7:S_4)$	$(C_3^{12}.C_2^6.C_2)$ . $S_4$	$C_3^6$ . $(C_3^6.C_2^7:S_4)$ (2)	$(C_3^{12}.C_2^6.C_2^3)$ . $S_3$	all 58

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{9}$
Commutator length:	$1$

Subgroups

There are 123 normal subgroups (51 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 $C_1$
Commutator:	a subgroup isomorphic to $C_3^{12}.C_2^6.C_3$
Frattini:	a subgroup isomorphic to $C_1$
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 90699264) are shown, as well as all normal subgroups of any index.

Order 1632586752: $C_3^6.(C_3^6.C_2^7:S_4)$

Order 816293376: $C_3^6.(C_3^6.C_2^6:S_4)$ x 2, $C_3^6.(C_3^6.C_2^6:S_4)$ x 2, $C_3^6.(C_3^6.C_2^6:S_4)$ x 2, $C_3^6.(C_6^3.S_3^3:S_4)$ x 2, $C_3^6.(C_3^6.C_2^6:S_4)$, $C_3^6.(C_3^6.C_2^7:A_4)$, $C_3^6.(C_3^6.C_2^6:S_4)$, $C_3^6.(C_3^6.C_2^6:S_4)$, $C_3^6.(C_3^6.C_2^6:S_4)$, $C_3^6.(C_3^6.C_2^6:S_4)$, $C_3^6.(C_3^6.C_2^6:S_4)$

Order 544195584: $C_3^{12}.C_2^5.C_2^5$

Order 408146688: $C_3^{12}.C_2^6.D_6$ x 2, $C_3^{12}.C_2^6.D_6$ x 2, $C_3^{12}.C_2^6.D_6$ x 2, $C_3^{12}.C_2^6.D_6$ x 2, $C_3^{12}.C_2^6.D_6$ x 2, $C_3^{12}.C_2^6.D_6$ x 2, $C_3^{12}.C_2^6.D_6$ x 2, $C_3^{12}.C_2^6.D_6$ x 2, $C_3^{12}.C_2^6.D_6$ x 2, $C_3^{12}.C_2^6.D_6$ x 2, $C_3^{12}.C_2^6.C_6.C_2$ x 2, $C_3^{12}.C_2^6.C_6.C_2$ x 2, $C_3^{12}.C_2^6.D_6$ x 2, $C_3^{12}.C_2^6.D_6$ x 2, $C_3^{12}.C_2^4.C_6.C_2^3$, $C_3^{12}.C_2^6.D_6$, $C_3^{12}.C_2^6.D_6$, $C_3^{12}.C_2^6.D_6$, $C_3^{12}.C_2^6.C_6.C_2$, $C_3^{12}.C_2^6.C_6.C_2$, $C_3^{12}.C_2^6.D_6$, $C_3^{12}.C_2^6.C_6.C_2$

Order 272097792: $C_3^{12}.C_2^5.C_2^4$ x 19, $C_3^{12}.C_2^6.C_2^3$ x 10, $C_3^{12}.C_2^4.C_2^5$ x 2

Order 204073344: $C_3^{12}.C_2^4.C_6.C_2^2$ x 14, $C_3^6.(C_6^3.S_3\wr S_3)$ x 2, $C_3^8.C_6^3:(S_3\times S_4)$ x 2, $C_3^{12}.C_2^6.S_3$ x 2, $C_3^{12}.C_2^6.S_3$ x 2, $C_3^{12}.C_2^6.C_6$ x 2, $C_3^{12}.C_2^6.C_6$ x 2, $C_3^{12}.C_2^6.S_3$ x 2, $C_3^{12}.C_2^6.C_6$ x 2, $C_3^{12}.C_2^6.S_3$, $C_3^{12}.C_2^6.S_3$, $C_3^{12}.C_2^6.C_6$, $C_3^6.(S_3^3.S_3\wr S_3)$, $C_3^6.(C_3^6.C_2^4:S_4)$, $C_3^6.(C_3^6.C_2^4:S_4)$

Order 136048896: $C_3^{12}.C_2^5.C_2^3$ x 176, $C_3^{12}.C_2^6.C_2^2$ x 32, $C_3^{12}.C_2^4.C_2^4$ x 28, $C_3^9.C_6^3.C_2^5$ x 19, $C_3^{12}.C_2^3.C_2^5$ x 2, $C_3^{11}.C_2^4.C_6.C_2^3$, $C_3^8.C_6^3.C_6.C_2^4$

Order 102036672: $C_3^{12}.C_2^4.D_6$ x 24, $C_3^{12}.C_2^4.C_6.C_2$ x 6, $C_3^9.C_6^2.D_6^2$ x 2, $C_3^{12}.C_2^4.D_6$ x 2, $C_3^{11}.C_2^4.S_3^2$ x 2, $C_3^{12}.C_2^4.D_6$ x 2, $C_3^{11}.C_2^4.S_3^2$ x 2, $C_3^{12}.C_2^4.D_6$ x 2, $C_3^8.(C_3\times C_6^3:S_4)$ x 2, $C_3^{12}.C_2^4.D_6$ x 2, $C_3^8.(C_3\times C_6^3):S_4$ x 2, $C_3^{11}.C_2^4.S_3^2$ x 2, $C_3^{11}.C_2^4.S_3^2$ x 2, $C_3^{12}.C_2^4.D_6$ x 2, $C_3^6.(C_6^3.S_3\wr C_3)$ x 2, $C_3^8.C_6^3:(S_3\times A_4)$ x 2, $C_3^{12}.C_2^4.D_6$ x 2, $C_3^{12}.C_2^4.D_6$ x 2, $C_3^{12}.C_2^4.D_6$ x 2, $C_3^{12}.C_2^4.D_6$ x 2, $C_3^{12}.C_2^4.D_6$ x 2, $C_3^{12}.C_2^4.D_6$ x 2, $C_3^{12}.C_2^4.D_6$ x 2, $C_3^{12}.A_4.C_2^4$, $C_3^{12}.C_2^6.C_3$, $C_3^{12}.C_2^4.D_6$, $C_3^{12}.C_2^4.D_6$, $C_3^{12}.C_2^4.D_6$, $C_3^{12}.C_2^4.D_6$, $C_3^6.(S_3^3.S_3\wr C_3)$, $C_3^{12}.C_2^4.C_6.C_2$, $C_3^{12}.C_2^4.C_6.C_2$

Order 90699264: $C_3^{11}.C_2^6.C_2^3$ x 3, $C_3^8.C_6^3.C_2^6$

Order 68024448: $C_3^{12}.C_2^5.C_2^2$ x 4, $C_3^{12}.C_2^6.C_2$ x 2, $C_3^{12}.C_2^4.C_2^3$, $C_3^{12}.C_2^3.C_2^4$

Order 34012224: $C_3^{12}.C_2^6$ x 8

Order 17006112: $C_3^{12}.C_2^5$ x 7

Order 8503056: $C_3^{12}.C_2^4$ x 4

Order 4251528: $C_3^{12}.C_2^3$ x 7, $C_3^9.C_6^3$ x 2

Order 2125764: $C_3^{12}.C_2^2$ x 2, $C_3^{11}.D_6$

Order 1062882: $C_3^{12}.C_2$

Order 531441: $C_3^{12}$

Order 5832: $C_3^6:C_2^3$ x 2

Order 2916: $C_3^5:D_6$ x 2

Order 729: $C_3^6$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 1632586752: $C_3^6.(C_3^6.C_2^7:S_4)$ ($C_1$)

Order 816293376: $C_3^6.(C_3^6.C_2^6:S_4)$ ($C_2$) x 2, $C_3^6.(C_3^6.C_2^6:S_4)$ ($C_2$) x 2, $C_3^6.(C_3^6.C_2^6:S_4)$ ($C_2$) x 2, $C_3^6.(C_6^3.S_3^3:S_4)$ ($C_2$) x 2, $C_3^6.(C_3^6.C_2^6:S_4)$ ($C_2$), $C_3^6.(C_3^6.C_2^7:A_4)$ ($C_2$), $C_3^6.(C_3^6.C_2^6:S_4)$ ($C_2$), $C_3^6.(C_3^6.C_2^6:S_4)$ ($C_2$), $C_3^6.(C_3^6.C_2^6:S_4)$ ($C_2$), $C_3^6.(C_3^6.C_2^6:S_4)$ ($C_2$), $C_3^6.(C_3^6.C_2^6:S_4)$ ($C_2$)

Order 408146688: $C_3^{12}.C_2^6.D_6$ ($C_2^2$) x 2, $C_3^{12}.C_2^6.D_6$ ($C_2^2$) x 2, $C_3^{12}.C_2^6.D_6$ ($C_2^2$) x 2, $C_3^{12}.C_2^6.D_6$ ($C_2^2$) x 2, $C_3^{12}.C_2^6.D_6$ ($C_2^2$) x 2, $C_3^{12}.C_2^6.D_6$ ($C_2^2$) x 2, $C_3^{12}.C_2^6.D_6$ ($C_2^2$) x 2, $C_3^{12}.C_2^6.D_6$ ($C_2^2$) x 2, $C_3^{12}.C_2^6.D_6$ ($C_2^2$) x 2, $C_3^{12}.C_2^6.D_6$ ($C_2^2$) x 2, $C_3^{12}.C_2^6.C_6.C_2$ ($C_2^2$) x 2, $C_3^{12}.C_2^6.C_6.C_2$ ($C_2^2$) x 2, $C_3^{12}.C_2^6.D_6$ ($C_2^2$) x 2, $C_3^{12}.C_2^6.D_6$ ($C_2^2$) x 2, $C_3^{12}.C_2^6.D_6$ ($C_2^2$), $C_3^{12}.C_2^6.D_6$ ($C_2^2$), $C_3^{12}.C_2^6.D_6$ ($C_2^2$), $C_3^{12}.C_2^6.C_6.C_2$ ($C_2^2$), $C_3^{12}.C_2^6.C_6.C_2$ ($C_2^2$), $C_3^{12}.C_2^6.D_6$ ($C_2^2$), $C_3^{12}.C_2^6.C_6.C_2$ ($C_2^2$)

Order 272097792: $C_3^{12}.C_2^6.C_2^3$ ($S_3$)

Order 204073344: $C_3^{12}.C_2^6.S_3$ ($C_2^3$) x 2, $C_3^{12}.C_2^6.S_3$ ($C_2^3$) x 2, $C_3^{12}.C_2^6.C_6$ ($C_2^3$) x 2, $C_3^{12}.C_2^6.C_6$ ($C_2^3$) x 2, $C_3^{12}.C_2^6.S_3$ ($C_2^3$) x 2, $C_3^{12}.C_2^6.C_6$ ($C_2^3$) x 2, $C_3^{12}.C_2^6.S_3$ ($C_2^3$), $C_3^{12}.C_2^6.S_3$ ($C_2^3$), $C_3^{12}.C_2^6.C_6$ ($C_2^3$)

Order 136048896: $C_3^{12}.C_2^6.C_2^2$ ($D_6$) x 6, $C_3^{12}.C_2^5.C_2^3$ ($D_6$)

Order 102036672: $C_3^{12}.C_2^6.C_3$ ($C_2^4$)

Order 68024448: $C_3^{12}.C_2^5.C_2^2$ ($C_2\times D_6$) x 4, $C_3^{12}.C_2^6.C_2$ ($C_2\times D_6$), $C_3^{12}.C_2^6.C_2$ ($S_4$), $C_3^{12}.C_2^4.C_2^3$ ($C_2\times D_6$), $C_3^{12}.C_2^3.C_2^4$ ($C_2\times D_6$)

Order 34012224: $C_3^{12}.C_2^6$ ($C_2\times S_4$) x 7, $C_3^{12}.C_2^6$ ($C_2^2\times D_6$)

Order 17006112: $C_3^{12}.C_2^5$ ($C_2^2\times S_4$) x 7

Order 8503056: $C_3^{12}.C_2^4$ ($C_2^3:S_4$) x 3, $C_3^{12}.C_2^4$ ($C_2^3\times S_4$)

Order 4251528: $C_3^{12}.C_2^3$ ($C_2^4:S_4$) x 7, $C_3^9.C_6^3$ ($C_2^4:S_4$) x 2

Order 2125764: $C_3^{12}.C_2^2$ ($C_2^5:S_4$) x 2, $C_3^{11}.D_6$ ($C_2^5:S_4$)

Order 1062882: $C_3^{12}.C_2$ ($C_2^6:S_4$)

Order 531441: $C_3^{12}$ ($C_2^7:S_4$)

Order 5832: $C_3^6:C_2^3$ ($C_3^6.C_2^4:S_4$) x 2

Order 2916: $C_3^5:D_6$ ($C_3^6.C_2^5:S_4$) x 2

Order 729: $C_3^6$ ($C_3^6.C_2^7:S_4$) x 2

Order 1: $C_1$ ($C_3^6.(C_3^6.C_2^7:S_4)$)

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 $3170 \times 3170$ character table is not available for this group.

Rational character table

The $3168 \times 3168$ rational character table is not available for this group.