Abstract group 8957952.v: $C_3^6.(C_2^9.S_4)$

• Label: 8957952.v
• Order: \( 2^{12} \cdot 3^{7} \)
• Exponent: \( 2^{3} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{18} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: not computed
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^6.(C_2^9.S_4)$
Order:	\(8957952\)\(\medspace = 2^{12} \cdot 3^{7} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Automorphism group:	Group of order \(573308928\)\(\medspace = 2^{18} \cdot 3^{7} \)
Composition factors:	$C_2$ x 12, $C_3$ x 7
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	9	12	18	24
Elements	1	51903	42200	772416	1362600	248832	331776	3286656	1990656	870912	8957952
Conjugacy classes	1	63	14	68	602	4	5	280	23	12	1072
Divisions	1	63	14	68	542	4	5	236	23	8	964
Autjugacy classes	1	29	14	21	279	2	3	91	7	4	451

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	12	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k \mid c^{2}=e^{2}=g^{6}=h^{6}=i^{12}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,8,21,6,10,20)(2,7,22,5,9,19)(3,12,23)(4,11,24)(13,34,26,14,33,25)(15,32,29,16,31,30) \!\cdots\! \rangle$
Transitive group:	36T58551				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:	$C_6^6$ . $(C_2^3:S_4)$ (2)	$C_3^6$ . $(C_2^9.S_4)$	$(C_3^5:D_6)$ . $(C_2^7:S_4)$	$(C_3^6.(C_2^8.S_4))$ . $C_2$	all 40

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{9}$
Commutator length:	$1$

Subgroups

There are 82 normal subgroups (38 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3^6.C_2^6.S_4.C_2^2$
Commutator:	$G' \simeq$ $C_3^6.C_2^5.C_2^4.C_3$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_3^6.C_2^6:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_6^6$	$G/\operatorname{Fit} \simeq$ $C_2^3:S_4$
Radical:	$R \simeq$ $C_3^6.(C_2^9.S_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^3\times C_6^3$	$G/\operatorname{soc} \simeq$ $C_2^6:S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2.C_2^6.C_2^5$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2\wr C_3$

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 4 (order at least 2239488) are shown, as well as all normal subgroups of any index.

Order 8957952: $C_3^6.(C_2^9.S_4)$

Order 4478976: $C_3^6.(C_2^8.S_4)$ x 4, $C_3^6.(C_2^9.A_4)$, $C_3^6.(C_2^8.S_4)$, $C_3^6.(C_2^8.S_4)$

Order 2985984: $C_3^6.C_2^5.C_2^6.C_2$

Order 2239488: $C_3^6.C_2^7:S_4$ x 2, $C_3^6.C_2^7:S_4$ x 2, $C_3^6.(C_2^8.A_4)$ x 2, $C_3^6.C_2^6.C_6.C_2^3$, $C_3^6.(C_2^7.S_4)$, $C_3^6.C_2^7:S_4$, $C_3^6.C_2^8:A_4$

Order 1492992: $C_3^6.C_2^6.C_2^5$

Order 1119744: $C_3^6.C_2^5.C_2^4.C_3$

Order 746496: $C_3^6.C_2^6.C_2^4$ x 2, $C_3^6.C_2^6.C_2^4$

Order 373248: $C_3^6.C_2^6.C_2^3$, $C_3^6.C_2^6.C_2^3$, $C_3^6.C_2^6.C_2^3$, $C_2^2\times C_3^6.C_2^6.C_2$

Order 186624: $C_2^2\times C_3^4.D_6\wr C_2.C_2$ x 2, $C_2^2\times C_3^6.C_2^6$ x 2, $C_3^6.C_2^6.C_2^2$ x 2, $C_3^6.C_2^5.C_2^3$, $C_2^2\times C_3^6.C_2^6$, $C_3^6.C_2^6.C_2^2$

Order 93312: $C_2^2\times C_3^6.C_2^5$, $C_3^6.C_2^5.C_2^2$, $C_2^2\times C_3^6.C_2^5$, $C_2^2\times C_3^6.C_2^5$

Order 46656: $C_2\times C_3^6.C_2^5$ x 4, $C_2^2\times C_3^6.C_2^4$ x 2, $C_2^2\times C_3^6.C_2^4$ x 2, $C_6^6$

Order 23328: $C_2\times C_3^6.C_2^4$ x 4, $C_2^2\times C_3^6.C_2^3$ x 2, $C_2\times C_3^6.C_2^4$ x 2, $C_3\times C_6^5$

Order 12288: $C_2^9.S_4$

Order 11664: $C_3^6.C_2^4$ x 2, $C_2\times C_3^6.C_2^3$, $C_3^2\times C_6^4$

Order 5832: $C_3^6:C_2^3$ x 4, $C_3^3\times C_6^3$ x 3, $C_3^6.C_2^3$ x 2

Order 4096: $C_2.C_2^6.C_2^5$

Order 2916: $C_3^5:D_6$ x 2, $C_3^4\times C_6^2$

Order 2187: $C_3^2\wr C_3$

Order 1458: $C_3^5\times C_6$

Order 729: $C_3^6$

Order 64: $C_2^6$

Order 32: $C_2^5$

Order 16: $C_2^4$

Order 8: $C_2^3$ x 3

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 4 (order at least 2239488) are shown, as well as all normal subgroups of any index.

Order 8957952: $C_3^6.(C_2^9.S_4)$

Order 4478976: $C_3^6.(C_2^9.A_4)$, $C_3^6.(C_2^8.S_4)$, $C_3^6.(C_2^8.S_4)$, $C_3^6.(C_2^8.S_4)$

Order 2985984: $C_3^6.C_2^5.C_2^6.C_2$

Order 2239488: $C_3^6.C_2^6.C_6.C_2^3$, $C_3^6.(C_2^7.S_4)$, $C_3^6.C_2^7:S_4$, $C_3^6.C_2^7:S_4$, $C_3^6.C_2^7:S_4$, $C_3^6.(C_2^8.A_4)$, $C_3^6.C_2^8:A_4$

Order 1492992: $C_3^6.C_2^6.C_2^5$

Order 1119744: $C_3^6.C_2^5.C_2^4.C_3$

Order 746496: $C_3^6.C_2^6.C_2^4$, $C_3^6.C_2^6.C_2^4$

Order 373248: $C_3^6.C_2^6.C_2^3$, $C_3^6.C_2^6.C_2^3$, $C_3^6.C_2^6.C_2^3$, $C_2^2\times C_3^6.C_2^6.C_2$

Order 186624: $C_3^6.C_2^5.C_2^3$, $C_2^2\times C_3^4.D_6\wr C_2.C_2$, $C_2^2\times C_3^6.C_2^6$, $C_3^6.C_2^6.C_2^2$, $C_2^2\times C_3^6.C_2^6$, $C_3^6.C_2^6.C_2^2$

Order 93312: $C_2^2\times C_3^6.C_2^5$, $C_3^6.C_2^5.C_2^2$, $C_2^2\times C_3^6.C_2^5$, $C_2^2\times C_3^6.C_2^5$

Order 46656: $C_2^2\times C_3^6.C_2^4$ x 2, $C_2^2\times C_3^6.C_2^4$, $C_2\times C_3^6.C_2^5$, $C_6^6$

Order 23328: $C_2^2\times C_3^6.C_2^3$ x 2, $C_2\times C_3^6.C_2^4$, $C_3\times C_6^5$, $C_2\times C_3^6.C_2^4$

Order 12288: $C_2^9.S_4$

Order 11664: $C_2\times C_3^6.C_2^3$, $C_3^6.C_2^4$, $C_3^2\times C_6^4$

Order 5832: $C_3^3\times C_6^3$ x 2, $C_3^6.C_2^3$, $C_3^6:C_2^3$

Order 4096: $C_2.C_2^6.C_2^5$

Order 2916: $C_3^4\times C_6^2$, $C_3^5:D_6$

Order 2187: $C_3^2\wr C_3$

Order 1458: $C_3^5\times C_6$

Order 729: $C_3^6$

Order 64: $C_2^6$

Order 32: $C_2^5$

Order 16: $C_2^4$

Order 8: $C_2^3$ x 2

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 8957952: $C_3^6.(C_2^9.S_4)$ ($C_1$)

Order 4478976: $C_3^6.(C_2^8.S_4)$ ($C_2$) x 4, $C_3^6.(C_2^9.A_4)$ ($C_2$), $C_3^6.(C_2^8.S_4)$ ($C_2$), $C_3^6.(C_2^8.S_4)$ ($C_2$)

Order 2239488: $C_3^6.C_2^7:S_4$ ($C_2^2$) x 2, $C_3^6.C_2^7:S_4$ ($C_2^2$) x 2, $C_3^6.(C_2^8.A_4)$ ($C_2^2$) x 2, $C_3^6.C_2^8:A_4$ ($C_2^2$)

Order 1492992: $C_3^6.C_2^6.C_2^5$ ($S_3$)

Order 1119744: $C_3^6.C_2^5.C_2^4.C_3$ ($C_2^3$)

Order 746496: $C_3^6.C_2^6.C_2^4$ ($D_6$) x 2, $C_3^6.C_2^6.C_2^4$ ($D_6$)

Order 373248: $C_3^6.C_2^6.C_2^3$ ($S_4$), $C_3^6.C_2^6.C_2^3$ ($S_4$), $C_3^6.C_2^6.C_2^3$ ($C_2\times D_6$), $C_2^2\times C_3^6.C_2^6.C_2$ ($S_4$)

Order 186624: $C_2^2\times C_3^4.D_6\wr C_2.C_2$ ($C_2\times S_4$) x 2, $C_2^2\times C_3^6.C_2^6$ ($C_2\times S_4$) x 2, $C_3^6.C_2^6.C_2^2$ ($C_2\times S_4$) x 2, $C_3^6.C_2^5.C_2^3$ ($C_2\times S_4$), $C_2^2\times C_3^6.C_2^6$ ($C_2\times S_4$), $C_3^6.C_2^6.C_2^2$ ($C_2\times S_4$)

Order 93312: $C_2^2\times C_3^6.C_2^5$ ($C_2^2\times S_4$), $C_3^6.C_2^5.C_2^2$ ($C_2^2\times S_4$), $C_2^2\times C_3^6.C_2^5$ ($C_2^2\times S_4$), $C_2^2\times C_3^6.C_2^5$ ($C_2^2:S_4$)

Order 46656: $C_2\times C_3^6.C_2^5$ ($C_2^3:S_4$) x 4, $C_2^2\times C_3^6.C_2^4$ ($C_2^3:S_4$) x 2, $C_2^2\times C_3^6.C_2^4$ ($C_2^3:S_4$), $C_2^2\times C_3^6.C_2^4$ ($C_2^3:S_4$), $C_6^6$ ($C_2^3:S_4$)

Order 23328: $C_2\times C_3^6.C_2^4$ ($C_2^4:S_4$) x 4, $C_2\times C_3^6.C_2^4$ ($C_2^4:S_4$) x 2, $C_3\times C_6^5$ ($C_2^4:S_4$), $C_2^2\times C_3^6.C_2^3$ ($C_2^2\wr S_3$), $C_2^2\times C_3^6.C_2^3$ ($C_2^4:S_4$)

Order 11664: $C_3^6.C_2^4$ ($C_2^5:S_4$) x 2, $C_2\times C_3^6.C_2^3$ ($C_2^5:S_4$), $C_3^2\times C_6^4$ ($C_2^5:S_4$)

Order 5832: $C_3^6:C_2^3$ ($C_2\wr S_4$) x 4, $C_3^3\times C_6^3$ ($C_2^6:S_4$) x 2, $C_3^6.C_2^3$ ($C_2^6:S_4$) x 2, $C_3^3\times C_6^3$ ($C_2^6:S_4$)

Order 2916: $C_3^5:D_6$ ($C_2^7:S_4$) x 2, $C_3^4\times C_6^2$ ($C_2^5.S_4.C_2^2$)

Order 1458: $C_3^5\times C_6$ ($C_2^6.A_4.C_2^3$)

Order 729: $C_3^6$ ($C_2^9.S_4$)

Order 64: $C_2^6$ ($C_3^6:C_2^3:S_4$)

Order 32: $C_2^5$ ($C_3^6.C_2^4:S_4$)

Order 16: $C_2^4$ ($C_3^6.C_2^5:S_4$)

Order 8: $C_2^3$ ($C_3^6.C_2^5.S_4.C_2$) x 2, $C_2^3$ ($C_3^6.C_2^6:S_4$)

Order 4: $C_2^2$ ($C_3^6.C_2^5.A_4.C_2^3$)

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

Order 1: $C_1$ ($C_3^6.(C_2^9.S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 8957952: $C_3^6.(C_2^9.S_4)$ ($C_1$)

Order 4478976: $C_3^6.(C_2^9.A_4)$ ($C_2$), $C_3^6.(C_2^8.S_4)$ ($C_2$), $C_3^6.(C_2^8.S_4)$ ($C_2$), $C_3^6.(C_2^8.S_4)$ ($C_2$)

Order 2239488: $C_3^6.C_2^7:S_4$ ($C_2^2$), $C_3^6.C_2^7:S_4$ ($C_2^2$), $C_3^6.(C_2^8.A_4)$ ($C_2^2$), $C_3^6.C_2^8:A_4$ ($C_2^2$)

Order 1492992: $C_3^6.C_2^6.C_2^5$ ($S_3$)

Order 1119744: $C_3^6.C_2^5.C_2^4.C_3$ ($C_2^3$)

Order 746496: $C_3^6.C_2^6.C_2^4$ ($D_6$), $C_3^6.C_2^6.C_2^4$ ($D_6$)

Order 373248: $C_3^6.C_2^6.C_2^3$ ($S_4$), $C_3^6.C_2^6.C_2^3$ ($S_4$), $C_3^6.C_2^6.C_2^3$ ($C_2\times D_6$), $C_2^2\times C_3^6.C_2^6.C_2$ ($S_4$)

Order 186624: $C_3^6.C_2^5.C_2^3$ ($C_2\times S_4$), $C_2^2\times C_3^4.D_6\wr C_2.C_2$ ($C_2\times S_4$), $C_2^2\times C_3^6.C_2^6$ ($C_2\times S_4$), $C_3^6.C_2^6.C_2^2$ ($C_2\times S_4$), $C_2^2\times C_3^6.C_2^6$ ($C_2\times S_4$), $C_3^6.C_2^6.C_2^2$ ($C_2\times S_4$)

Order 93312: $C_2^2\times C_3^6.C_2^5$ ($C_2^2\times S_4$), $C_3^6.C_2^5.C_2^2$ ($C_2^2\times S_4$), $C_2^2\times C_3^6.C_2^5$ ($C_2^2\times S_4$), $C_2^2\times C_3^6.C_2^5$ ($C_2^2:S_4$)

Order 46656: $C_2^2\times C_3^6.C_2^4$ ($C_2^3:S_4$), $C_2^2\times C_3^6.C_2^4$ ($C_2^3:S_4$), $C_2^2\times C_3^6.C_2^4$ ($C_2^3:S_4$), $C_2\times C_3^6.C_2^5$ ($C_2^3:S_4$), $C_6^6$ ($C_2^3:S_4$)

Order 23328: $C_2\times C_3^6.C_2^4$ ($C_2^4:S_4$), $C_3\times C_6^5$ ($C_2^4:S_4$), $C_2^2\times C_3^6.C_2^3$ ($C_2^2\wr S_3$), $C_2^2\times C_3^6.C_2^3$ ($C_2^4:S_4$), $C_2\times C_3^6.C_2^4$ ($C_2^4:S_4$)

Order 11664: $C_2\times C_3^6.C_2^3$ ($C_2^5:S_4$), $C_3^6.C_2^4$ ($C_2^5:S_4$), $C_3^2\times C_6^4$ ($C_2^5:S_4$)

Order 5832: $C_3^3\times C_6^3$ ($C_2^6:S_4$), $C_3^3\times C_6^3$ ($C_2^6:S_4$), $C_3^6.C_2^3$ ($C_2^6:S_4$), $C_3^6:C_2^3$ ($C_2\wr S_4$)

Order 2916: $C_3^4\times C_6^2$ ($C_2^5.S_4.C_2^2$), $C_3^5:D_6$ ($C_2^7:S_4$)

Order 1458: $C_3^5\times C_6$ ($C_2^6.A_4.C_2^3$)

Order 729: $C_3^6$ ($C_2^9.S_4$)

Order 64: $C_2^6$ ($C_3^6:C_2^3:S_4$)

Order 32: $C_2^5$ ($C_3^6.C_2^4:S_4$)

Order 16: $C_2^4$ ($C_3^6.C_2^5:S_4$)

Order 8: $C_2^3$ ($C_3^6.C_2^5.S_4.C_2$), $C_2^3$ ($C_3^6.C_2^6:S_4$)

Order 4: $C_2^2$ ($C_3^6.C_2^5.A_4.C_2^3$)

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

Order 1: $C_1$ ($C_3^6.(C_2^9.S_4)$)

Series

Derived series

$C_3^6.(C_2^9.S_4)$

$\rhd$

$C_3^6.C_2^5.C_2^4.C_3$

$\rhd$

$C_3^6.C_2^6.C_2^3$

$\rhd$

$C_3^3\times C_6^3$

$\rhd$

$C_1$

Chief series

$C_3^6.(C_2^9.S_4)$

$\rhd$

$C_3^6.(C_2^9.A_4)$

$\rhd$

$C_3^6.(C_2^8.A_4)$

$\rhd$

$C_3^6.C_2^5.C_2^4.C_3$

$\rhd$

$C_3^6.C_2^6.C_2^3$

$\rhd$

$C_2^2\times C_3^6.C_2^5$

$\rhd$

$C_2^2\times C_3^6.C_2^3$

$\rhd$

$C_3^3\times C_6^3$

$\rhd$

$C_3^4\times C_6^2$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Lower central series

$C_3^6.(C_2^9.S_4)$

$\rhd$

$C_3^6.C_2^5.C_2^4.C_3$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

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

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

Character theory

Complex character table

The $1072 \times 1072$ character table is not available for this group.

Rational character table

The $964 \times 964$ rational character table is not available for this group.