Abstract group 864.2236: $C_3^2:(C_4\times S_4)$

• Label: 864.2236
• Order: \( 2^{5} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $17$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_3^2:(C_4\times S_4)$
Order:	\(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^2\times C_6^2:D_6$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Composition factors:	$C_2$ x 5, $C_3$ x 3
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12
Elements	1	43	80	180	200	360	864
Conjugacy classes	1	5	4	10	12	8	40
Divisions	1	5	4	6	12	5	33
Autjugacy classes	1	4	4	5	11	4	29

Dimension	1	2	3	4	6	12
Irr. complex chars.	8	8	8	2	12	2	40
Irr. rational chars.	4	6	4	4	12	3	33

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	$54$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	12
Arbitrary	6	8	8

Constructions

Presentation:	$\langle a, b, c, d \mid b^{12}=c^{6}=d^{6}=[a,d]=[c,d]=1, a^{2}=b^{6}, b^{a}=b^{5}, c^{a}=c^{5}d^{3}, c^{b}=c^{5}d^{3}, d^{b}=cd^{2} \rangle$
Permutation group:	Degree $17$ $\langle(3,7)(5,6)(8,9)(14,15,16,17), (2,4)(3,7)(5,8)(6,9)(11,12)(14,15,16,17), (14,16) \!\cdots\! \rangle$
Transitive group:	36T1337				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6.S_4)$ $\,\rtimes\,$ $S_3$	$(C_3^2:S_4)$ $\,\rtimes\,$ $C_4$ (2)	$(C_6^2:C_4)$ $\,\rtimes\,$ $S_3$	$(C_3^2:C_4)$ $\,\rtimes\,$ $S_4$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(S_3\times S_4)$	$(C_6\times A_4)$ . $D_6$	$(C_6^2:D_6)$ . $C_2$	$(C_2\times C_6^2)$ . $D_6$	all 11

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

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

Subgroups

There are 1994 subgroups in 235 conjugacy classes, 25 normal (23 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_6^2:D_6$
Commutator:	$G' \simeq$ $C_3^2:A_4$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_6$	$G/\Phi \simeq$ $S_3\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6^2$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_3^2:(C_4\times S_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 864: $C_3^2:(C_4\times S_4)$

Order 432: $C_6^2:D_6$, $C_6^2:C_{12}$, $(C_3\times A_4):C_{12}$

Order 288: $C_2^3.S_3^2$, $C_2^3.S_3^2$

Order 216: $C_3^2:S_4$ x 2, $C_6.S_3^2$, $C_6^2:C_6$

Order 144: $C_6\times S_4$ x 2, $C_6^2:C_4$, $C_6.D_{12}$, $C_6.D_{12}$, $C_3^2:C_4^2$, $C_6^2:C_4$, $C_6^2:C_2^2$, $D_6.D_6$, $A_4\times C_3:C_4$, $C_6.S_4$

Order 108: $C_3^2:C_{12}$ x 2, $C_3^2:D_6$, $C_3^2:A_4$

Order 96: $C_2^3.D_6$, $C_4\times S_4$, $C_2^3.D_6$

Order 72: $C_3\times S_4$ x 4, $C_6\wr C_2$ x 4, $C_6:C_{12}$ x 3, $C_6.D_6$ x 3, $C_6\times A_4$ x 2, $C_6.D_6$ x 2, $C_2\times C_6^2$, $C_6\times D_6$

Order 54: $C_3^2:S_3$ x 2, $C_2\times \He_3$

Order 48: $C_6.C_2^3$ x 3, $C_2\times S_4$ x 2, $C_6.D_4$ x 2, $C_{12}:C_4$ x 2, $C_6\times D_4$, $C_6:D_4$, $C_4\times D_6$, $C_4\times A_4$, $A_4:C_4$, $C_2^2:C_{12}$, $D_6:C_4$, $C_{12}:C_4$, $C_6.D_4$

Order 36: $C_6\times S_3$ x 6, $C_3:C_{12}$ x 5, $C_3^2:C_4$ x 3, $C_6^2$ x 3, $C_3\times A_4$ x 2

Order 32: $C_4\times D_4$

Order 27: $\He_3$

Order 24: $C_6:C_4$ x 11, $C_3:D_4$ x 4, $S_4$ x 4, $C_3\times D_4$ x 4, $C_2\times C_{12}$ x 3, $C_4\times S_3$ x 3, $C_2^2\times C_6$ x 3, $C_2\times A_4$ x 2, $C_2\times D_6$

Order 18: $C_3\times S_3$ x 6, $C_3\times C_6$ x 5

Order 16: $C_2^2:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_4:C_4$, $C_4^2$, $C_2\times D_4$

Order 12: $C_2\times C_6$ x 12, $C_3:C_4$ x 10, $D_6$ x 6, $C_{12}$ x 5, $A_4$ x 2

Order 9: $C_3^2$ x 3

Order 8: $C_2\times C_4$ x 7, $D_4$ x 4, $C_2^3$ x 2

Order 6: $C_6$ x 12, $S_3$ x 6

Order 4: $C_2^2$ x 7, $C_4$ x 6

Order 3: $C_3$ x 4

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 864: $C_3^2:(C_4\times S_4)$

Order 432: $C_6^2:D_6$, $C_6^2:C_{12}$, $(C_3\times A_4):C_{12}$

Order 288: $C_2^3.S_3^2$, $C_2^3.S_3^2$

Order 216: $C_3^2:S_4$, $C_6.S_3^2$, $C_6^2:C_6$

Order 144: $C_6\times S_4$ x 2, $C_6^2:C_4$, $C_6.D_{12}$, $C_6.D_{12}$, $C_3^2:C_4^2$, $C_6^2:C_4$, $C_6^2:C_2^2$, $D_6.D_6$, $A_4\times C_3:C_4$, $C_6.S_4$

Order 108: $C_3^2:C_{12}$ x 2, $C_3^2:D_6$, $C_3^2:A_4$

Order 96: $C_2^3.D_6$, $C_4\times S_4$, $C_2^3.D_6$

Order 72: $C_6:C_{12}$ x 3, $C_6.D_6$ x 3, $C_6\times A_4$ x 2, $C_3\times S_4$ x 2, $C_6.D_6$ x 2, $C_6\wr C_2$ x 2, $C_2\times C_6^2$, $C_6\times D_6$

Order 54: $C_3^2:S_3$, $C_2\times \He_3$

Order 48: $C_6.C_2^3$ x 3, $C_2\times S_4$ x 2, $C_6.D_4$ x 2, $C_{12}:C_4$ x 2, $C_6\times D_4$, $C_6:D_4$, $C_4\times D_6$, $C_4\times A_4$, $A_4:C_4$, $C_2^2:C_{12}$, $D_6:C_4$, $C_{12}:C_4$, $C_6.D_4$

Order 36: $C_6\times S_3$ x 5, $C_3:C_{12}$ x 4, $C_3^2:C_4$ x 3, $C_6^2$ x 3, $C_3\times A_4$ x 2

Order 32: $C_4\times D_4$

Order 27: $\He_3$

Order 24: $C_6:C_4$ x 11, $C_2\times C_{12}$ x 3, $C_4\times S_3$ x 3, $C_2^2\times C_6$ x 3, $C_3:D_4$ x 2, $C_2\times A_4$ x 2, $S_4$ x 2, $C_3\times D_4$ x 2, $C_2\times D_6$

Order 18: $C_3\times C_6$ x 5, $C_3\times S_3$ x 3

Order 16: $C_2^2:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_4:C_4$, $C_4^2$, $C_2\times D_4$

Order 12: $C_2\times C_6$ x 11, $C_3:C_4$ x 9, $D_6$ x 5, $C_{12}$ x 4, $A_4$ x 2

Order 9: $C_3^2$ x 3

Order 8: $C_2\times C_4$ x 7, $C_2^3$ x 2, $D_4$ x 2

Order 6: $C_6$ x 11, $S_3$ x 3

Order 4: $C_2^2$ x 6, $C_4$ x 5

Order 3: $C_3$ x 4

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 864: $C_3^2:(C_4\times S_4)$ ($C_1$)

Order 432: $C_6^2:D_6$ ($C_2$), $C_6^2:C_{12}$ ($C_2$), $(C_3\times A_4):C_{12}$ ($C_2$)

Order 216: $C_3^2:S_4$ ($C_4$) x 2, $C_6^2:C_6$ ($C_2^2$)

Order 144: $C_6^2:C_4$ ($S_3$), $C_6.S_4$ ($S_3$)

Order 108: $C_3^2:A_4$ ($C_2\times C_4$)

Order 72: $C_2\times C_6^2$ ($D_6$), $C_6\times A_4$ ($D_6$)

Order 36: $C_3^2:C_4$ ($S_4$), $C_6^2$ ($C_4\times S_3$), $C_3\times A_4$ ($C_4\times S_3$)

Order 24: $C_2^2\times C_6$ ($S_3^2$)

Order 18: $C_3\times C_6$ ($C_2\times S_4$)

Order 12: $C_2\times C_6$ ($C_6.D_6$)

Order 9: $C_3^2$ ($C_4\times S_4$)

Order 8: $C_2^3$ ($C_3^2:D_6$)

Order 6: $C_6$ ($S_3\times S_4$)

Order 4: $C_2^2$ ($C_6.S_3^2$)

Order 3: $C_3$ ($C_2^3.S_3^2$)

Order 2: $C_2$ ($C_6^2:D_6$)

Order 1: $C_1$ ($C_3^2:(C_4\times S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 864: $C_3^2:(C_4\times S_4)$ ($C_1$)

Order 432: $C_6^2:D_6$ ($C_2$), $C_6^2:C_{12}$ ($C_2$), $(C_3\times A_4):C_{12}$ ($C_2$)

Order 216: $C_3^2:S_4$ ($C_4$), $C_6^2:C_6$ ($C_2^2$)

Order 144: $C_6^2:C_4$ ($S_3$), $C_6.S_4$ ($S_3$)

Order 108: $C_3^2:A_4$ ($C_2\times C_4$)

Order 72: $C_2\times C_6^2$ ($D_6$), $C_6\times A_4$ ($D_6$)

Order 36: $C_3^2:C_4$ ($S_4$), $C_6^2$ ($C_4\times S_3$), $C_3\times A_4$ ($C_4\times S_3$)

Order 24: $C_2^2\times C_6$ ($S_3^2$)

Order 18: $C_3\times C_6$ ($C_2\times S_4$)

Order 12: $C_2\times C_6$ ($C_6.D_6$)

Order 9: $C_3^2$ ($C_4\times S_4$)

Order 8: $C_2^3$ ($C_3^2:D_6$)

Order 6: $C_6$ ($S_3\times S_4$)

Order 4: $C_2^2$ ($C_6.S_3^2$)

Order 3: $C_3$ ($C_2^3.S_3^2$)

Order 2: $C_2$ ($C_6^2:D_6$)

Order 1: $C_1$ ($C_3^2:(C_4\times S_4)$)

Series

Derived series

$C_3^2:(C_4\times S_4)$

$\rhd$

$C_3^2:A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_1$

Chief series

$C_3^2:(C_4\times S_4)$

$\rhd$

$(C_3\times A_4):C_{12}$

$\rhd$

$C_6^2:C_6$

$\rhd$

$C_3^2:A_4$

$\rhd$

$C_6^2$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_3^2:(C_4\times S_4)$

$\rhd$

$C_3^2:A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 17 larger groups in the database.

This group is a maximal quotient of 21 larger groups in the database.

Character theory

Complex character table

See the $40 \times 40$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $33 \times 33$ rational character table.