Abstract group 432.594: $C_2.S_3^3$

• Label: 432.594
• Order: \( 2^{4} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $13$
• Trans deg.: $48$
• Rank: $3$

Group information

Description:	$C_2.S_3^3$
Order:	\(432\)\(\medspace = 2^{4} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$D_6\times D_6\wr C_2$, of order \(3456\)\(\medspace = 2^{7} \cdot 3^{3} \)
Composition factors:	$C_2$ x 4, $C_3$ 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	31	26	96	158	120	432
Conjugacy classes	1	7	7	8	21	10	54
Divisions	1	7	7	4	21	5	45
Autjugacy classes	1	3	5	3	9	3	24

Dimension	1	2	4	8
Irr. complex chars.	16	24	12	2	54
Irr. rational chars.	8	20	14	3	45

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$48$
Rank:	$3$
Inequivalent generating triples:	$5376$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	16	16
Arbitrary	6	8	8

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{6}=c^{12}=d^{3}=[a,c]=[a,d]=[b,d]=1, b^{a}=b^{5}, c^{b}=c^{5}, d^{c}=d^{2} \rangle$
Permutation group:	Degree $13$ $\langle(1,2)(3,5)(4,6), (1,2)(3,6)(4,5)(10,11)(12,13), (8,9)(10,12,11,13), (10,11)(12,13), (7,8,9), (1,3,4)(2,5,6), (1,4,3)(2,5,6)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 10 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 15 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 15 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 18 & 23 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 6 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/30\Z)$
Direct product:	$S_3$ ${}^2$ $\, \times\, $ $(C_3:C_4)$
Semidirect product:	$C_3$ $\,\rtimes\,$ $(D_6.D_6)$ (2)	$(C_3:C_{12})$ $\,\rtimes\,$ $D_6$ (2)	$(C_6.S_3^2)$ $\,\rtimes\,$ $C_2$ (2)	$(C_6.S_3^2)$ $\,\rtimes\,$ $C_2$ (2)	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_6$ . $S_3^2$ (2)	$C_2$ . $S_3^3$	$(S_3\times D_6)$ . $S_3$	$(C_6\times S_3)$ . $D_6$ (4)	all 11

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

Homology

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

Subgroups

There are 1244 subgroups in 270 conjugacy classes, 74 normal (18 characteristic).

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

Special subgroups

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

Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

Order 432: $C_2.S_3^3$

Order 216: $C_6.S_3^2$ x 2, $C_6.S_3^2$ x 2, $C_6\times S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$

Order 144: $D_6.D_6$ x 2, $C_4\times S_3^2$

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

Order 72: $C_6.D_6$ x 13, $C_6\times D_6$ x 2, $C_6.D_6$ x 2, $C_6:C_{12}$ x 2, $S_3\times C_{12}$ x 2, $S_3\times D_6$, $C_{12}:S_3$, $C_6.D_6$

Order 54: $S_3\times C_3^2$ x 4, $C_3^2:C_6$ x 2, $C_3^2\times C_6$

Order 48: $C_4\times D_6$ x 2, $C_6.C_2^3$

Order 36: $C_6\times S_3$ x 17, $C_3^2:C_4$ x 9, $C_3:C_{12}$ x 9, $S_3^2$ x 4, $C_6^2$ x 2, $C_3\times C_{12}$, $C_6:S_3$

Order 27: $C_3^3$

Order 24: $C_6:C_4$ x 10, $C_4\times S_3$ x 9, $C_2\times C_{12}$ x 2, $C_2\times D_6$ x 2, $C_2^2\times C_6$

Order 18: $C_3\times S_3$ x 18, $C_3\times C_6$ x 11, $C_3:S_3$ x 2

Order 16: $C_2^2\times C_4$

Order 12: $C_3:C_4$ x 14, $D_6$ x 13, $C_2\times C_6$ x 11, $C_{12}$ x 5

Order 9: $C_3^2$ x 7

Order 8: $C_2\times C_4$ x 6, $C_2^3$

Order 6: $C_6$ x 21, $S_3$ x 10

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

Order 3: $C_3$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 432: $C_2.S_3^3$

Order 216: $C_6\times S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6.S_3^2$

Order 144: $D_6.D_6$, $C_4\times S_3^2$

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

Order 72: $C_6.D_6$ x 7, $C_6\times D_6$, $S_3\times D_6$, $C_6.D_6$, $C_{12}:S_3$, $C_6:C_{12}$, $S_3\times C_{12}$, $C_6.D_6$

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

Order 48: $C_6.C_2^3$, $C_4\times D_6$

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

Order 27: $C_3^3$

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

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

Order 16: $C_2^2\times C_4$

Order 12: $C_3:C_4$ x 9, $C_2\times C_6$ x 5, $D_6$ x 4, $C_{12}$ x 3

Order 9: $C_3^2$ x 5

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

Order 6: $C_6$ x 9, $S_3$ x 3

Order 4: $C_2^2$ x 3, $C_4$ x 3

Order 3: $C_3$ x 5

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 432: $C_2.S_3^3$ ($C_1$)

Order 216: $C_6.S_3^2$ ($C_2$) x 2, $C_6.S_3^2$ ($C_2$) x 2, $C_6\times S_3^2$ ($C_2$), $C_6.S_3^2$ ($C_2$), $C_6.S_3^2$ ($C_2$)

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

Order 72: $C_6.D_6$ ($S_3$) x 2, $S_3\times D_6$ ($S_3$)

Order 54: $S_3\times C_3^2$ ($C_2\times C_4$) x 4, $C_3^2:C_6$ ($C_2\times C_4$) x 2, $C_3^2\times C_6$ ($C_2^3$)

Order 36: $C_6\times S_3$ ($D_6$) x 4, $S_3^2$ ($C_3:C_4$) x 4, $C_3^2:C_4$ ($D_6$) x 2, $C_3:C_{12}$ ($D_6$) x 2, $C_6:S_3$ ($D_6$)

Order 27: $C_3^3$ ($C_2^2\times C_4$)

Order 18: $C_3\times S_3$ ($C_6:C_4$) x 4, $C_3\times S_3$ ($C_4\times S_3$) x 4, $C_3\times C_6$ ($C_2\times D_6$) x 3, $C_3:S_3$ ($C_6:C_4$) x 2

Order 12: $D_6$ ($S_3^2$) x 2, $C_3:C_4$ ($S_3^2$)

Order 9: $C_3^2$ ($C_4\times D_6$) x 2, $C_3^2$ ($C_6.C_2^3$)

Order 6: $S_3$ ($C_6.D_6$) x 4, $C_6$ ($S_3\times D_6$) x 3

Order 3: $C_3$ ($D_6.D_6$) x 2, $C_3$ ($C_4\times S_3^2$)

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

Order 1: $C_1$ ($C_2.S_3^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 432: $C_2.S_3^3$ ($C_1$)

Order 216: $C_6\times S_3^2$ ($C_2$), $C_6.S_3^2$ ($C_2$), $C_6.S_3^2$ ($C_2$), $C_6.S_3^2$ ($C_2$), $C_6.S_3^2$ ($C_2$)

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

Order 72: $S_3\times D_6$ ($S_3$), $C_6.D_6$ ($S_3$)

Order 54: $C_3^2\times C_6$ ($C_2^3$), $C_3^2:C_6$ ($C_2\times C_4$), $S_3\times C_3^2$ ($C_2\times C_4$)

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

Order 27: $C_3^3$ ($C_2^2\times C_4$)

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

Order 12: $D_6$ ($S_3^2$), $C_3:C_4$ ($S_3^2$)

Order 9: $C_3^2$ ($C_6.C_2^3$), $C_3^2$ ($C_4\times D_6$)

Order 6: $C_6$ ($S_3\times D_6$) x 2, $S_3$ ($C_6.D_6$)

Order 3: $C_3$ ($D_6.D_6$), $C_3$ ($C_4\times S_3^2$)

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

Order 1: $C_1$ ($C_2.S_3^3$)

Series

Derived series

$C_2.S_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Chief series

$C_2.S_3^3$

$\rhd$

$C_6.S_3^2$

$\rhd$

$C_3^2:C_{12}$

$\rhd$

$C_3:C_{12}$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2.S_3^3$

$\rhd$

$C_3^3$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 39 larger groups in the database.

This group is a maximal quotient of 42 larger groups in the database.

Character theory

Complex character table

See the $54 \times 54$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $45 \times 45$ rational character table.