Abstract group 864.4367: $C_4\times S_3^3$

• Label: 864.4367
• Order: \( 2^{5} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $13$
• Trans deg.: $48$
• Rank: $4$

Group information

Description:	$C_4\times S_3^3$
Order:	\(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^2\times S_3^3:S_4$, of order \(20736\)\(\medspace = 2^{8} \cdot 3^{4} \)
Composition factors:	$C_2$ x 5, $C_3$ x 3
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.

Group statistics

Order	1	2	3	4	6	12
Elements	1	127	26	128	278	304	864
Conjugacy classes	1	15	7	16	31	38	108
Divisions	1	15	7	8	31	19	81
Autjugacy classes	1	4	3	4	6	6	24

Dimension	1	2	4	8	16
Irr. complex chars.	32	48	24	4	0	108
Irr. rational chars.	16	32	24	8	1	81

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$48$
Rank:	$4$
Inequivalent generating quadruples:	$7381920$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	16	16
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{6}=c^{6}=d^{12}=[a,c]=[a,d]=[b,d]=1, b^{a}=b^{5}, c^{b}=c^{5}, d^{c}=d^{5} \rangle$
Permutation group:	Degree $13$ $\langle(8,9), (1,2)(3,5)(4,6), (1,2)(3,6)(4,5), (10,11,12,13), (10,12)(11,13), (1,3,4)(2,5,6), (1,4,3)(2,5,6), (7,8,9)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 19 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 15 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 20 \\ 15 & 7 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 6 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/30\Z)$
Direct product:	$C_4$ $\, \times\, $ $S_3$ ${}^3$
Semidirect product:	$(C_6.D_6)$ $\,\rtimes\,$ $D_6$	$(C_6.D_6)$ $\,\rtimes\,$ $D_6$	$(S_3\times C_{12})$ $\,\rtimes\,$ $D_6$	$(C_{12}:S_3)$ $\,\rtimes\,$ $D_6$	all 39
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $D_6^2$	$(S_3\times D_6)$ . $D_6$	$D_6$ . $(S_3\times D_6)$	$(C_2\times S_3^3)$ . $C_2$	all 15

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

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 6928 subgroups in 1122 conjugacy classes, 226 normal (14 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $S_3^3$
Commutator:	$G' \simeq$ $C_3^3$	$G/G' \simeq$ $C_2^3\times C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times S_3^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2\times C_{12}$	$G/\operatorname{Fit} \simeq$ $C_2^3$
Radical:	$R \simeq$ $C_4\times S_3^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3\times C_4$
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: $C_4\times S_3^3$

Order 432: $C_{12}:S_3^2$ x 3, $C_{12}\times S_3^2$ x 3, $C_2.S_3^3$ x 3, $C_2.S_3^3$ x 3, $C_2\times S_3^3$, $C_{12}:S_3^2$, $C_2.S_3^3$

Order 288: $C_2.D_6^2$ x 3

Order 216: $S_3^3$ x 8, $C_6.S_3^2$ x 6, $C_6:S_3^2$ x 3, $C_6\times S_3^2$ x 3, $C_3:S_3\times C_{12}$ x 3, $C_6.C_6^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6:S_3^2$, $C_4\times C_3^2:S_3$

Order 144: $C_4\times S_3^2$ x 27, $C_{12}\times D_6$ x 6, $D_6.D_6$ x 6, $D_6^2$ x 3, $C_{12}:D_6$ x 3, $C_2^2.S_3^2$ x 3

Order 108: $C_3:S_3^2$ x 12, $C_3\times S_3^2$ x 12, $C_3:S_3^2$ x 4, $C_3^2:D_6$ x 3, $C_3^2\times D_6$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:D_6$, $C_3^2\times C_{12}$, $C_3^3:C_4$

Order 96: $C_{12}:C_2^3$ x 3

Order 72: $S_3\times D_6$ x 39, $S_3\times C_{12}$ x 30, $C_6.D_6$ x 30, $C_{12}:S_3$ x 16, $C_6.D_6$ x 15, $C_6\times D_6$ x 6, $C_6:C_{12}$ x 6, $C_6:D_6$ x 3, $C_6\times C_{12}$ x 3, $C_6.D_6$ x 3

Order 54: $C_3^2:C_6$ x 6, $S_3\times C_3^2$ x 6, $C_3^2:S_3$ x 2, $C_3^2\times C_6$

Order 48: $C_4\times D_6$ x 39, $C_2^2\times D_6$ x 3, $C_2^2\times C_{12}$ x 3, $C_6.C_2^3$ x 3

Order 36: $S_3^2$ x 60, $C_6\times S_3$ x 42, $C_6:S_3$ x 22, $C_3:C_{12}$ x 18, $C_3\times C_{12}$ x 10, $C_3^2:C_4$ x 10, $C_6^2$ x 3

Order 32: $C_2^3\times C_4$

Order 27: $C_3^3$

Order 24: $C_4\times S_3$ x 61, $C_2\times D_6$ x 45, $C_2\times C_{12}$ x 21, $C_6:C_4$ x 21, $C_2^2\times C_6$ x 3

Order 18: $C_3\times S_3$ x 36, $C_3:S_3$ x 20, $C_3\times C_6$ x 13

Order 16: $C_2^2\times C_4$ x 14, $C_2^4$

Order 12: $D_6$ x 103, $C_2\times C_6$ x 24, $C_{12}$ x 19, $C_3:C_4$ x 19

Order 9: $C_3^2$ x 7

Order 8: $C_2\times C_4$ x 28, $C_2^3$ x 15

Order 6: $S_3$ x 38, $C_6$ x 31

Order 4: $C_2^2$ x 35, $C_4$ x 8

Order 3: $C_3$ x 7

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 864: $C_4\times S_3^3$

Order 432: $C_2\times S_3^3$, $C_{12}:S_3^2$, $C_{12}:S_3^2$, $C_{12}\times S_3^2$, $C_2.S_3^3$, $C_2.S_3^3$, $C_2.S_3^3$

Order 288: $C_2.D_6^2$

Order 216: $C_6:S_3^2$, $C_6:S_3^2$, $C_6\times S_3^2$, $S_3^3$, $C_4\times C_3^2:S_3$, $C_3:S_3\times C_{12}$, $C_6.C_6^2$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$

Order 144: $C_4\times S_3^2$ x 7, $D_6^2$, $C_{12}:D_6$, $C_{12}\times D_6$, $C_2^2.S_3^2$, $D_6.D_6$

Order 108: $C_3^2:D_6$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3:S_3^2$, $C_3:S_3^2$, $C_3\times S_3^2$, $C_3^2\times C_{12}$, $C_3^3:C_4$, $C_3^2:C_{12}$, $C_3^2:C_{12}$

Order 96: $C_{12}:C_2^3$

Order 72: $C_{12}:S_3$ x 6, $S_3\times C_{12}$ x 6, $C_6.D_6$ x 6, $S_3\times D_6$ x 5, $C_6.D_6$ x 4, $C_6:D_6$, $C_6\times D_6$, $C_6\times C_{12}$, $C_6.D_6$, $C_6:C_{12}$

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

Order 48: $C_4\times D_6$ x 9, $C_2^2\times D_6$, $C_2^2\times C_{12}$, $C_6.C_2^3$

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

Order 32: $C_2^3\times C_4$

Order 27: $C_3^3$

Order 24: $C_4\times S_3$ x 15, $C_2\times D_6$ x 6, $C_2\times C_{12}$ x 5, $C_6:C_4$ x 5, $C_2^2\times C_6$

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

Order 16: $C_2^2\times C_4$ x 6, $C_2^4$

Order 12: $D_6$ x 11, $C_{12}$ x 6, $C_3:C_4$ x 6, $C_2\times C_6$ x 4

Order 9: $C_3^2$ x 3

Order 8: $C_2\times C_4$ x 9, $C_2^3$ x 4

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

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 864: $C_4\times S_3^3$ ($C_1$)

Order 432: $C_{12}:S_3^2$ ($C_2$) x 3, $C_{12}\times S_3^2$ ($C_2$) x 3, $C_2.S_3^3$ ($C_2$) x 3, $C_2.S_3^3$ ($C_2$) x 3, $C_2\times S_3^3$ ($C_2$), $C_{12}:S_3^2$ ($C_2$), $C_2.S_3^3$ ($C_2$)

Order 216: $S_3^3$ ($C_4$) x 8, $C_6.S_3^2$ ($C_2^2$) x 6, $C_6:S_3^2$ ($C_2^2$) x 3, $C_6\times S_3^2$ ($C_2^2$) x 3, $C_3:S_3\times C_{12}$ ($C_2^2$) x 3, $C_6.C_6^2$ ($C_2^2$) x 3, $C_6.S_3^2$ ($C_2^2$) x 3, $C_6.S_3^2$ ($C_2^2$) x 3, $C_6.S_3^2$ ($C_2^2$) x 3, $C_6.S_3^2$ ($C_2^2$) x 3, $C_6.S_3^2$ ($C_2^2$) x 3, $C_6:S_3^2$ ($C_2^2$), $C_4\times C_3^2:S_3$ ($C_2^2$)

Order 144: $C_4\times S_3^2$ ($S_3$) x 3

Order 108: $C_3:S_3^2$ ($C_2\times C_4$) x 12, $C_3\times S_3^2$ ($C_2\times C_4$) x 12, $C_3:S_3^2$ ($C_2\times C_4$) x 4, $C_3^2:D_6$ ($C_2^3$) x 3, $C_3^2\times D_6$ ($C_2^3$) x 3, $C_3^2:C_{12}$ ($C_2^3$) x 3, $C_3^2:C_{12}$ ($C_2^3$) x 3, $C_3^2:D_6$ ($C_2^3$), $C_3^2\times C_{12}$ ($C_2^3$), $C_3^3:C_4$ ($C_2^3$)

Order 72: $S_3\times C_{12}$ ($D_6$) x 6, $C_6.D_6$ ($D_6$) x 6, $S_3\times D_6$ ($D_6$) x 3, $C_{12}:S_3$ ($D_6$) x 3, $C_6.D_6$ ($D_6$) x 3

Order 54: $C_3^2:C_6$ ($C_2^2\times C_4$) x 6, $S_3\times C_3^2$ ($C_2^2\times C_4$) x 6, $C_3^2:S_3$ ($C_2^2\times C_4$) x 2, $C_3^2\times C_6$ ($C_2^4$)

Order 36: $S_3^2$ ($C_4\times S_3$) x 12, $C_3:C_{12}$ ($C_2\times D_6$) x 6, $C_6\times S_3$ ($C_2\times D_6$) x 6, $C_3\times C_{12}$ ($C_2\times D_6$) x 3, $C_3^2:C_4$ ($C_2\times D_6$) x 3, $C_6:S_3$ ($C_2\times D_6$) x 3

Order 27: $C_3^3$ ($C_2^3\times C_4$)

Order 24: $C_4\times S_3$ ($S_3^2$) x 3

Order 18: $C_3\times S_3$ ($C_4\times D_6$) x 12, $C_3:S_3$ ($C_4\times D_6$) x 6, $C_3\times C_6$ ($C_2^2\times D_6$) x 3

Order 12: $D_6$ ($S_3\times D_6$) x 3, $C_{12}$ ($S_3\times D_6$) x 3, $C_3:C_4$ ($S_3\times D_6$) x 3

Order 9: $C_3^2$ ($C_{12}:C_2^3$) x 3

Order 6: $S_3$ ($C_4\times S_3^2$) x 6, $C_6$ ($D_6^2$) x 3

Order 4: $C_4$ ($S_3^3$)

Order 3: $C_3$ ($C_2.D_6^2$) x 3

Order 2: $C_2$ ($C_2\times S_3^3$)

Order 1: $C_1$ ($C_4\times S_3^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 864: $C_4\times S_3^3$ ($C_1$)

Order 432: $C_2\times S_3^3$ ($C_2$), $C_{12}:S_3^2$ ($C_2$), $C_{12}:S_3^2$ ($C_2$), $C_{12}\times S_3^2$ ($C_2$), $C_2.S_3^3$ ($C_2$), $C_2.S_3^3$ ($C_2$), $C_2.S_3^3$ ($C_2$)

Order 216: $C_6:S_3^2$ ($C_2^2$), $C_6:S_3^2$ ($C_2^2$), $C_6\times S_3^2$ ($C_2^2$), $S_3^3$ ($C_4$), $C_4\times C_3^2:S_3$ ($C_2^2$), $C_3:S_3\times C_{12}$ ($C_2^2$), $C_6.C_6^2$ ($C_2^2$), $C_6.S_3^2$ ($C_2^2$), $C_6.S_3^2$ ($C_2^2$), $C_6.S_3^2$ ($C_2^2$), $C_6.S_3^2$ ($C_2^2$), $C_6.S_3^2$ ($C_2^2$), $C_6.S_3^2$ ($C_2^2$)

Order 144: $C_4\times S_3^2$ ($S_3$)

Order 108: $C_3^2:D_6$ ($C_2^3$), $C_3^2:D_6$ ($C_2^3$), $C_3^2\times D_6$ ($C_2^3$), $C_3:S_3^2$ ($C_2\times C_4$), $C_3:S_3^2$ ($C_2\times C_4$), $C_3\times S_3^2$ ($C_2\times C_4$), $C_3^2\times C_{12}$ ($C_2^3$), $C_3^3:C_4$ ($C_2^3$), $C_3^2:C_{12}$ ($C_2^3$), $C_3^2:C_{12}$ ($C_2^3$)

Order 72: $S_3\times D_6$ ($D_6$), $C_{12}:S_3$ ($D_6$), $S_3\times C_{12}$ ($D_6$), $C_6.D_6$ ($D_6$), $C_6.D_6$ ($D_6$)

Order 54: $C_3^2\times C_6$ ($C_2^4$), $C_3^2:S_3$ ($C_2^2\times C_4$), $C_3^2:C_6$ ($C_2^2\times C_4$), $S_3\times C_3^2$ ($C_2^2\times C_4$)

Order 36: $C_3\times C_{12}$ ($C_2\times D_6$), $C_3^2:C_4$ ($C_2\times D_6$), $C_3:C_{12}$ ($C_2\times D_6$), $C_6:S_3$ ($C_2\times D_6$), $C_6\times S_3$ ($C_2\times D_6$), $S_3^2$ ($C_4\times S_3$)

Order 27: $C_3^3$ ($C_2^3\times C_4$)

Order 24: $C_4\times S_3$ ($S_3^2$)

Order 18: $C_3\times C_6$ ($C_2^2\times D_6$), $C_3:S_3$ ($C_4\times D_6$), $C_3\times S_3$ ($C_4\times D_6$)

Order 12: $D_6$ ($S_3\times D_6$), $C_{12}$ ($S_3\times D_6$), $C_3:C_4$ ($S_3\times D_6$)

Order 9: $C_3^2$ ($C_{12}:C_2^3$)

Order 6: $C_6$ ($D_6^2$), $S_3$ ($C_4\times S_3^2$)

Order 4: $C_4$ ($S_3^3$)

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

Order 2: $C_2$ ($C_2\times S_3^3$)

Order 1: $C_1$ ($C_4\times S_3^3$)

Series

Derived series

$C_4\times S_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Chief series

$C_4\times S_3^3$

$\rhd$

$C_{12}\times S_3^2$

$\rhd$

$C_6.C_6^2$

$\rhd$

$S_3\times C_{12}$

$\rhd$

$C_3\times C_{12}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_4\times S_3^3$

$\rhd$

$C_3^3$

Upper central series

$C_1$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 35 larger groups in the database.

This group is a maximal quotient of 144 larger groups in the database.

Character theory

Complex character table

See the $108 \times 108$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $81 \times 81$ rational character table.