Abstract group 864.4431: $C_4\times C_3^2:S_4$

• Label: 864.4431
• Order: \( 2^{5} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3^{3} \)
• Perm deg.: $14$
• Trans deg.: $108$
• Rank: $4$

Group information

Description:	$C_4\times C_3^2:S_4$
Order:	\(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^2\times C_3^4.Q_8.(S_3\times S_4)$, of order \(373248\)\(\medspace = 2^{9} \cdot 3^{6} \)
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	115	80	332	128	208	864
Conjugacy classes	1	5	13	10	21	34	84
Divisions	1	5	13	6	21	17	63
Autjugacy classes	1	4	2	5	4	3	19

Dimension	1	2	3	4	6	12
Irr. complex chars.	8	52	8	0	16	0	84
Irr. rational chars.	4	28	4	13	10	4	63

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$108$
Rank:	$4$
Inequivalent generating quadruples:	$687960$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{12}=c^{6}=d^{6}=[c,d]=1, b^{a}=b^{5}, c^{a}=c^{2}d^{3}, d^{a}=c^{3}d^{2}, c^{b}=c^{4}d^{3}, d^{b}=c^{3}d \rangle$
Permutation group:	Degree $14$ $\langle(2,3)(5,6)(12,13), (7,8,9,10), (7,9)(8,10), (1,2,3)(4,5,6)(12,13,14), (1,2,3), (4,5,6), (11,12)(13,14), (11,13)(12,14)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 13 & 12 \\ 36 & 1 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 49 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 30 & 13 \end{array}\right), \left(\begin{array}{rr} 43 & 30 \\ 30 & 43 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 45 & 31 \end{array}\right), \left(\begin{array}{rr} 53 & 52 \\ 51 & 7 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/60\Z)$
Direct product:	$C_4$ $\, \times\, $ $(C_3^2:S_4)$
Semidirect product:	$(C_{12}\times A_4)$ $\,\rtimes\,$ $S_3$	$A_4$ $\,\rtimes\,$ $(C_{12}:S_3)$	$(C_3\times C_{12})$ $\,\rtimes\,$ $S_4$	$C_{12}$ $\,\rtimes\,$ $(C_3:S_4)$	all 17
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(C_6:S_4)$	$(C_6\times A_4)$ . $D_6$	$(C_6^2:D_6)$ . $C_2$	$(C_2\times C_6^2)$ . $D_6$	all 10

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:	$C_{3} \times C_{6}^{2}$
Commutator length:	$1$

Subgroups

There are 5752 subgroups in 592 conjugacy classes, 107 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $C_3^2:S_4$
Commutator:	$G' \simeq$ $C_3^2\times A_4$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_6^2:D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6\times C_{12}$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_4\times C_3^2:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6^2$	$G/\operatorname{soc} \simeq$ $D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

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

Order 432: $C_6^2:D_6$, $C_6^2:C_{12}$, $(C_3^2\times A_4):C_4$

Order 288: $C_{12}:S_4$ x 12, $C_6^2.C_2^3$

Order 216: $C_3^2:S_4$ x 2, $C_6^2:C_6$, $C_4\times C_3^2:S_3$

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

Order 108: $C_3^2:D_6$, $C_3^2\times A_4$, $C_3^2\times C_{12}$, $C_3^3:C_4$

Order 96: $C_4\times S_4$ x 9, $C_{12}:D_4$ x 4

Order 72: $C_3:S_4$ x 24, $C_{12}:S_3$ x 14, $C_6\times A_4$ x 12, $C_6^2:C_2$ x 4, $C_6.D_6$ x 3, $C_6\times C_{12}$ x 2, $C_2\times C_6^2$, $C_6:D_6$

Order 54: $C_3^2:S_3$ x 2, $C_3^2\times C_6$

Order 48: $C_2\times S_4$ x 9, $C_4\times A_4$ x 9, $A_4:C_4$ x 9, $C_2^2\times C_{12}$ x 4, $C_6:D_4$ x 4, $C_4\times D_6$ x 4, $C_6.D_4$ x 4, $D_6:C_4$ x 4, $C_6.D_4$ x 4, $C_{12}:C_4$ x 4

Order 36: $C_3^2:C_4$ x 16, $C_6:S_3$ x 16, $C_3\times C_{12}$ x 14, $C_3\times A_4$ x 12, $C_6^2$ x 3

Order 32: $C_4\times D_4$

Order 27: $C_3^3$

Order 24: $S_4$ x 18, $C_4\times S_3$ x 17, $C_3:D_4$ x 16, $C_6:C_4$ x 12, $C_2\times A_4$ x 9, $C_2\times C_{12}$ x 8, $C_2^2\times C_6$ x 4, $C_2\times D_6$ x 4

Order 18: $C_3:S_3$ x 26, $C_3\times C_6$ x 15

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: $D_6$ x 25, $C_3:C_4$ x 25, $C_{12}$ x 17, $C_2\times C_6$ x 12, $A_4$ x 9

Order 9: $C_3^2$ x 13

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

Order 6: $S_3$ x 26, $C_6$ x 21

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

Order 3: $C_3$ x 13

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

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

Order 432: $C_6^2:D_6$, $C_6^2:C_{12}$, $(C_3^2\times A_4):C_4$

Order 288: $C_{12}:S_4$, $C_6^2.C_2^3$

Order 216: $C_6^2:C_6$, $C_3^2:S_4$, $C_4\times C_3^2:S_3$

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

Order 108: $C_3^2:D_6$, $C_3^2\times A_4$, $C_3^2\times C_{12}$, $C_3^3:C_4$

Order 96: $C_4\times S_4$, $C_{12}:D_4$

Order 72: $C_6.D_6$ x 3, $C_{12}:S_3$ x 3, $C_6\times C_{12}$ x 2, $C_6^2:C_2$ x 2, $C_2\times C_6^2$, $C_6:D_6$, $C_6\times A_4$, $C_3:S_4$

Order 54: $C_3^2\times C_6$, $C_3^2:S_3$

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

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

Order 32: $C_4\times D_4$

Order 27: $C_3^3$

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

Order 18: $C_3\times C_6$ x 4, $C_3:S_3$ x 2

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: $D_6$ x 4, $C_3:C_4$ x 4, $C_2\times C_6$ x 3, $C_{12}$ x 3, $A_4$

Order 9: $C_3^2$ x 2

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

Order 6: $C_6$ x 4, $S_3$ x 2

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 432: $C_6^2:D_6$ ($C_2$), $C_6^2:C_{12}$ ($C_2$), $(C_3^2\times A_4):C_4$ ($C_2$)

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

Order 144: $C_{12}\times A_4$ ($S_3$) x 12, $C_2\times C_6\times C_{12}$ ($S_3$)

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

Order 72: $C_6\times A_4$ ($D_6$) x 12, $C_2\times C_6^2$ ($D_6$)

Order 48: $C_4\times A_4$ ($C_3:S_3$) x 9, $C_2^2\times C_{12}$ ($C_3:S_3$) x 4

Order 36: $C_3\times A_4$ ($C_4\times S_3$) x 12, $C_3\times C_{12}$ ($S_4$), $C_6^2$ ($C_4\times S_3$)

Order 24: $C_2\times A_4$ ($C_6:S_3$) x 9, $C_2^2\times C_6$ ($C_6:S_3$) x 4

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

Order 16: $C_2^2\times C_4$ ($C_3^2:S_3$)

Order 12: $A_4$ ($C_{12}:S_3$) x 9, $C_2\times C_6$ ($C_{12}:S_3$) x 4, $C_{12}$ ($C_3:S_4$) x 4

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

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

Order 6: $C_6$ ($C_6:S_4$) x 4

Order 4: $C_2^2$ ($C_4\times C_3^2:S_3$), $C_4$ ($C_3^2:S_4$)

Order 3: $C_3$ ($C_{12}:S_4$) x 4

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

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

Normal subgroups up to automorphism (quotient in parentheses)

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

Order 432: $C_6^2:D_6$ ($C_2$), $C_6^2:C_{12}$ ($C_2$), $(C_3^2\times A_4):C_4$ ($C_2$)

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

Order 144: $C_2\times C_6\times C_{12}$ ($S_3$), $C_{12}\times A_4$ ($S_3$)

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

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

Order 48: $C_2^2\times C_{12}$ ($C_3:S_3$), $C_4\times A_4$ ($C_3:S_3$)

Order 36: $C_3\times C_{12}$ ($S_4$), $C_6^2$ ($C_4\times S_3$), $C_3\times A_4$ ($C_4\times S_3$)

Order 24: $C_2^2\times C_6$ ($C_6:S_3$), $C_2\times A_4$ ($C_6:S_3$)

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

Order 16: $C_2^2\times C_4$ ($C_3^2:S_3$)

Order 12: $C_2\times C_6$ ($C_{12}:S_3$), $A_4$ ($C_{12}:S_3$), $C_{12}$ ($C_3:S_4$)

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

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

Order 6: $C_6$ ($C_6:S_4$)

Order 4: $C_2^2$ ($C_4\times C_3^2:S_3$), $C_4$ ($C_3^2:S_4$)

Order 3: $C_3$ ($C_{12}:S_4$)

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

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

Series

Derived series

$C_4\times C_3^2:S_4$

$\rhd$

$C_3^2\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_4\times C_3^2:S_4$

$\rhd$

$C_6^2:D_6$

$\rhd$

$C_6^2:C_6$

$\rhd$

$C_3^2\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_4\times C_3^2:S_4$

$\rhd$

$C_3^2\times A_4$

Upper central series

$C_1$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 27 larger groups in the database.

This group is a maximal quotient of 14 larger groups in the database.

Character theory

Complex character table

See the $84 \times 84$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $63 \times 63$ rational character table.