Abstract group 2239488.dh: $C_3^6.C_2^7:S_4$

• Label: 2239488.dh
• Order: \( 2^{10} \cdot 3^{7} \)
• Exponent: \( 2^{3} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $30$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^6.C_2^7:S_4$
Order:	\(2239488\)\(\medspace = 2^{10} \cdot 3^{7} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Automorphism group:	$C_2\times C_3^6.C_2^4.C_2^4.C_6.C_2^5$, of order \(71663616\)\(\medspace = 2^{15} \cdot 3^{7} \)
Composition factors:	$C_2$ x 10, $C_3$ 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	14719	42200	207360	259304	93312	331776	772416	331776	186624	2239488
Conjugacy classes	1	23	21	24	397	4	8	150	8	4	640
Divisions	1	23	17	24	307	2	5	124	5	2	510
Autjugacy classes	1	16	14	10	166	1	3	46	3	1	261

Minimal presentations

Permutation degree:	$30$
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:	${\langle a, b, c, d, e, f, g, h, i, j \mid b^{3}=d^{6}=e^{6}=g^{12}=h^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $30$ $\langle(1,3,5)(2,6,13)(4,10,8)(7,11,12)(9,16,17)(14,15,18)(19,20,23)(21,25,27) \!\cdots\! \rangle$
Transitive group:	36T43744	36T43745			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:	$C_3^6$ . $(C_2^7:S_4)$	$C_2^5$ . $(C_3^5:D_6:S_4)$	$C_2$ . $(C_3^6.C_2^6.S_4)$	$(C_3^6.C_2^7:A_4)$ . $C_2$	all 15

Elements of the group are displayed as permutations of degree 30.

Homology

Abelianization:	$C_{2} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$2$

Subgroups

There are 31 normal subgroups (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3^6.C_2^6.S_4$
Commutator:	$G' \simeq$ $C_3^6.C_2^7:A_4$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_3^6.C_2^4:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6^5$	$G/\operatorname{Fit} \simeq$ $C_2^2:S_4$
Radical:	$R \simeq$ $C_3^6.C_2^7:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^3\times C_6^3$	$G/\operatorname{soc} \simeq$ $C_2^4:S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4.C_2^4.C_2^2$
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 8 (order at least 279936) are shown, as well as all normal subgroups of any index.

Order 2239488: $C_3^6.C_2^7:S_4$

Order 1119744: $C_3^6.C_2^7:A_4$

Order 746496: $(C_3^5\times C_6).C_2^5.C_2^4$

Order 559872: $C_3^6.C_2^5:S_4$ x 2, $C_3^6.C_2^5:S_4$ x 2, $C_3^6.C_2^5:S_4$, $C_3^6.(C_2^5.S_4)$, $C_3^6.C_2^5:S_4$

Order 373248: $(C_3^5\times C_6).C_2^5.C_2^3$ x 2, $C_3^6.C_2^4.C_2^4.C_2$ x 2, $(C_3^2\times C_6^4).C_2.C_2^4$ x 2, $(C_3^5\times C_6).C_2^5.C_2^3$ x 2, $C_3^6.C_2^4.C_2^5$, $C_3^6.C_2^6.C_2^3$, $(C_3^5\times C_6).C_2^5.C_2^3$, $(C_3^5\times C_6).C_2^5.C_2^3$, $C_3^6.C_2^4.C_2.C_2^4$, $C_3^6.C_2^4.C_2.C_2^4$, $(C_3\times C_6^5).C_2^4$

Order 279936: $C_3^6.C_4^2:S_4$ x 2, $C_3^6.C_2^4:S_4$ x 2, $C_3^6.C_2^5:A_4$ x 2, $C_3^6.C_2^5:A_4$ x 2, $C_3^6.C_2^5:A_4$ x 2, $C_3^6.C_2^4:S_4$ x 2, $(C_3\times C_6^5).A_4$ x 2, $C_3^6.C_2^6.C_6$, $C_3^6.C_2^5.A_4$, $C_3^6.C_2^5:A_4$, $C_3^6.(C_2^5.A_4)$, $C_3^6.C_2^5:A_4$, $C_2^5.C_3\wr A_4$

Order 93312: $C_3^6.C_2^5.C_2^2$ x 2, $(C_3\times C_6^5).C_2^2$ x 2, $C_2^4\times C_3^6.C_2^3$, $(C_3^3\times C_6^3).C_2^4$, $C_2^2\times C_3^6.C_2^5$

Order 23328: $C_2^2\times C_3^6.C_2^3$ x 4, $C_3^6.(C_2\times C_4^2)$ x 2, $C_3\times C_6^5$

Order 11664: $C_2\times C_3^6.C_2^3$

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

Order 3072: $C_2^7:S_4$

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 1024: $C_2^4.C_2^4.C_2^2$

Order 729: $C_3^6$

Order 32: $C_2^5$

Order 8: $C_2^3$

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 8 (order at least 279936) are shown, as well as all normal subgroups of any index.

Order 2239488: $C_3^6.C_2^7:S_4$

Order 1119744: $C_3^6.C_2^7:A_4$

Order 746496: $(C_3^5\times C_6).C_2^5.C_2^4$

Order 559872: $C_3^6.C_2^5:S_4$, $C_3^6.C_2^5:S_4$, $C_3^6.(C_2^5.S_4)$, $C_3^6.C_2^5:S_4$, $C_3^6.C_2^5:S_4$

Order 373248: $C_3^6.C_2^4.C_2^5$, $C_3^6.C_2^6.C_2^3$, $(C_3^5\times C_6).C_2^5.C_2^3$, $(C_3^5\times C_6).C_2^5.C_2^3$, $C_3^6.C_2^4.C_2^4.C_2$, $(C_3^5\times C_6).C_2^5.C_2^3$, $C_3^6.C_2^4.C_2.C_2^4$, $(C_3^2\times C_6^4).C_2.C_2^4$, $C_3^6.C_2^4.C_2.C_2^4$, $(C_3^5\times C_6).C_2^5.C_2^3$, $(C_3\times C_6^5).C_2^4$

Order 279936: $C_3^6.C_2^6.C_6$, $C_3^6.C_4^2:S_4$, $C_3^6.C_2^4:S_4$, $C_3^6.C_2^5:A_4$, $C_3^6.C_2^5:A_4$, $C_3^6.C_2^5.A_4$, $C_3^6.C_2^5:A_4$, $C_3^6.C_2^5:A_4$, $C_3^6.(C_2^5.A_4)$, $C_3^6.C_2^5:A_4$, $C_3^6.C_2^4:S_4$, $(C_3\times C_6^5).A_4$, $C_2^5.C_3\wr A_4$

Order 93312: $C_3^6.C_2^5.C_2^2$, $C_2^4\times C_3^6.C_2^3$, $(C_3^3\times C_6^3).C_2^4$, $C_2^2\times C_3^6.C_2^5$, $(C_3\times C_6^5).C_2^2$

Order 23328: $C_2^2\times C_3^6.C_2^3$ x 3, $C_3\times C_6^5$, $C_3^6.(C_2\times C_4^2)$

Order 11664: $C_2\times C_3^6.C_2^3$

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

Order 3072: $C_2^7:S_4$

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 1024: $C_2^4.C_2^4.C_2^2$

Order 729: $C_3^6$

Order 32: $C_2^5$

Order 8: $C_2^3$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2239488: $C_3^6.C_2^7:S_4$ ($C_1$)

Order 1119744: $C_3^6.C_2^7:A_4$ ($C_2$)

Order 373248: $(C_3\times C_6^5).C_2^4$ ($S_3$)

Order 93312: $C_3^6.C_2^5.C_2^2$ ($S_4$) x 2, $(C_3\times C_6^5).C_2^2$ ($S_4$) x 2, $C_2^4\times C_3^6.C_2^3$ ($S_4$), $(C_3^3\times C_6^3).C_2^4$ ($S_4$), $C_2^2\times C_3^6.C_2^5$ ($S_4$)

Order 23328: $C_2^2\times C_3^6.C_2^3$ ($C_2^2:S_4$) x 4, $C_3^6.(C_2\times C_4^2)$ ($C_2^2:S_4$) x 2, $C_3\times C_6^5$ ($C_2^2:S_4$)

Order 11664: $C_2\times C_3^6.C_2^3$ ($C_2^3:S_4$)

Order 5832: $C_3^6.C_2^3$ ($C_4^2:S_4$) x 2, $C_3^3\times C_6^3$ ($C_2^4:S_4$)

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

Order 1458: $C_3^5\times C_6$ ($C_2^6.S_4$)

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

Order 32: $C_2^5$ ($C_3^5:D_6:S_4$)

Order 8: $C_2^3$ ($C_3^6.C_2^4:S_4$)

Order 4: $C_2^2$ ($C_3^6.C_2^5.S_4$)

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

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 2239488: $C_3^6.C_2^7:S_4$ ($C_1$)

Order 1119744: $C_3^6.C_2^7:A_4$ ($C_2$)

Order 373248: $(C_3\times C_6^5).C_2^4$ ($S_3$)

Order 93312: $C_3^6.C_2^5.C_2^2$ ($S_4$), $C_2^4\times C_3^6.C_2^3$ ($S_4$), $(C_3^3\times C_6^3).C_2^4$ ($S_4$), $C_2^2\times C_3^6.C_2^5$ ($S_4$), $(C_3\times C_6^5).C_2^2$ ($S_4$)

Order 23328: $C_2^2\times C_3^6.C_2^3$ ($C_2^2:S_4$) x 3, $C_3\times C_6^5$ ($C_2^2:S_4$), $C_3^6.(C_2\times C_4^2)$ ($C_2^2:S_4$)

Order 11664: $C_2\times C_3^6.C_2^3$ ($C_2^3:S_4$)

Order 5832: $C_3^3\times C_6^3$ ($C_2^4:S_4$), $C_3^6.C_2^3$ ($C_4^2:S_4$)

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

Order 1458: $C_3^5\times C_6$ ($C_2^6.S_4$)

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

Order 32: $C_2^5$ ($C_3^5:D_6:S_4$)

Order 8: $C_2^3$ ($C_3^6.C_2^4:S_4$)

Order 4: $C_2^2$ ($C_3^6.C_2^5.S_4$)

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

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

Series

Derived series

$C_3^6.C_2^7:S_4$

$\rhd$

$C_3^6.C_2^7:A_4$

$\rhd$

$(C_3\times C_6^5).C_2^4$

$\rhd$

$C_3^3\times C_6^3$

$\rhd$

$C_1$

Chief series

$C_3^6.C_2^7:S_4$

$\rhd$

$C_3^6.C_2^7:A_4$

$\rhd$

$(C_3\times C_6^5).C_2^4$

$\rhd$

$(C_3\times C_6^5).C_2^2$

$\rhd$

$C_3\times C_6^5$

$\rhd$

$C_3^3\times C_6^3$

$\rhd$

$C_3^4\times C_6^2$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Lower central series

$C_3^6.C_2^7:S_4$

$\rhd$

$C_3^6.C_2^7:A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

The $640 \times 640$ character table is not available for this group.

Rational character table

The $510 \times 510$ rational character table is not available for this group.