Abstract group 576.3123: $C_6.(S_3\times C_4^2)$

• Label: 576.3123
• Order: \( 2^{6} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \)
• Perm deg.: $18$
• Trans deg.: $192$
• Rank: $3$

Group information

Description:	$C_6.(S_3\times C_4^2)$
Order:	\(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	Group of order \(18432\)\(\medspace = 2^{11} \cdot 3^{2} \)
Composition factors:	$C_2$ x 6, $C_3$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	7	8	248	56	256	576
Conjugacy classes	1	7	3	32	21	32	96
Divisions	1	7	3	18	21	18	68
Autjugacy classes	1	7	3	7	21	7	46

Dimension	1	2	4	8
Irr. complex chars.	32	40	24	0	96
Irr. rational chars.	8	28	28	4	68

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$192$
Rank:	$3$
Inequivalent generating triples:	$2688$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	6	8	8

Constructions

Presentation:	$\langle a, b, c, d \mid a^{4}=b^{12}=c^{4}=d^{3}=[a,c]=[a,d]=[b,d]=1, b^{a}=b^{5}c^{2}, c^{b}=c^{3}, d^{c}=d^{2} \rangle$
Permutation group:	Degree $18$ $\langle(4,5)(6,7)(8,9)(10,11)(12,13,14,15)(17,18), (2,3)(4,6,5,7)(8,10,9,11)(12,13,14,15) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 53 & 5 \\ 2 & 7 \end{array}\right), \left(\begin{array}{rr} 41 & 0 \\ 0 & 41 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 49 \end{array}\right), \left(\begin{array}{rr} 16 & 15 \\ 45 & 31 \end{array}\right), \left(\begin{array}{rr} 43 & 40 \\ 40 & 23 \end{array}\right), \left(\begin{array}{rr} 13 & 32 \\ 56 & 17 \end{array}\right), \left(\begin{array}{rr} 13 & 12 \\ 36 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 40 \\ 40 & 53 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/60\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6.D_{12})$ $\,\rtimes\,$ $C_4$ (4)	$(C_3:C_{12})$ $\,\rtimes\,$ $C_4^2$ (4)	$C_3$ $\,\rtimes\,$ $(C_{12}:C_4^2)$	$(C_3^2:C_4^2)$ $\,\rtimes\,$ $C_4$ (4)	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6.D_6)$ . $Q_8$ (2)	$(C_6.D_6)$ . $D_4$ (2)	$C_6$ . $(D_6.D_4)$ (2)	$C_6$ . $(D_{12}:C_4)$	all 65

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

Homology

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

Subgroups

There are 1428 subgroups in 442 conjugacy classes, 156 normal (88 characteristic).

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

Special subgroups

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

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 576: $C_6.(S_3\times C_4^2)$

Order 288: $C_2^3.S_3^2$ x 2, $C_6^2.C_2^3$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$

Order 192: $C_{12}:C_4^2$, $C_6.(C_4\times D_4)$

Order 144: $C_3^2:C_4^2$ x 6, $C_6^2.C_2^2$ x 4, $C_6.D_{12}$ x 4, $C_6^2:C_4$ x 4, $C_3^2:C_4^2$ x 2, $C_6^2:C_4$ x 2, $C_2\times C_6\times C_{12}$

Order 96: $C_6:C_4^2$ x 7, $C_6.C_4^2$ x 3, $C_2^3.D_6$ x 2, $C_6.C_4^2$, $C_2\times C_4:C_{12}$, $C_2^2.D_{12}$

Order 72: $C_6:C_{12}$ x 18, $C_6.D_6$ x 10, $C_6\times C_{12}$ x 4, $C_2\times C_6^2$

Order 64: $C_4:C_4^2$

Order 48: $C_{12}:C_4$ x 20, $C_6.C_2^3$ x 10, $C_6.D_4$ x 8, $C_2^2\times C_{12}$ x 7, $C_4:C_{12}$ x 4, $C_{12}:C_4$ x 4

Order 36: $C_3:C_{12}$ x 10, $C_6^2$ x 7, $C_3^2:C_4$ x 6, $C_3\times C_{12}$ x 2

Order 32: $C_2\times C_4^2$ x 3, $C_2^2.D_4$ x 2, $C_2.C_4^2$ x 2

Order 24: $C_6:C_4$ x 50, $C_2\times C_{12}$ x 32, $C_2^2\times C_6$ x 3

Order 18: $C_3\times C_6$ x 7

Order 16: $C_4:C_4$ x 8, $C_4^2$ x 8, $C_2^2\times C_4$ x 7

Order 12: $C_3:C_4$ x 30, $C_2\times C_6$ x 21, $C_{12}$ x 18

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 32, $C_2^3$

Order 6: $C_6$ x 21

Order 4: $C_4$ x 18, $C_2^2$ x 7

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 576: $C_6.(S_3\times C_4^2)$

Order 288: $C_2^3.S_3^2$ x 2, $C_6^2.C_2^3$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2.D_4$

Order 192: $C_{12}:C_4^2$, $C_6.(C_4\times D_4)$

Order 144: $C_6^2:C_4$ x 4, $C_3^2:C_4^2$ x 2, $C_6^2:C_4$ x 2, $C_3^2:C_4^2$, $C_6^2.C_2^2$, $C_6.D_{12}$, $C_2\times C_6\times C_{12}$

Order 96: $C_6:C_4^2$ x 7, $C_6.C_4^2$ x 3, $C_2^3.D_6$ x 2, $C_6.C_4^2$, $C_2\times C_4:C_{12}$, $C_2^2.D_{12}$

Order 72: $C_6:C_{12}$ x 12, $C_6.D_6$ x 6, $C_6\times C_{12}$ x 3, $C_2\times C_6^2$

Order 64: $C_4:C_4^2$

Order 48: $C_6.C_2^3$ x 10, $C_2^2\times C_{12}$ x 7, $C_{12}:C_4$ x 7, $C_6.D_4$ x 2, $C_4:C_{12}$, $C_{12}:C_4$

Order 36: $C_6^2$ x 7, $C_3:C_{12}$ x 4, $C_3^2:C_4$ x 2, $C_3\times C_{12}$

Order 32: $C_2\times C_4^2$ x 3, $C_2^2.D_4$ x 2, $C_2.C_4^2$ x 2

Order 24: $C_6:C_4$ x 30, $C_2\times C_{12}$ x 21, $C_2^2\times C_6$ x 3

Order 18: $C_3\times C_6$ x 7

Order 16: $C_2^2\times C_4$ x 7, $C_4^2$ x 3, $C_4:C_4$ x 2

Order 12: $C_2\times C_6$ x 21, $C_3:C_4$ x 10, $C_{12}$ x 7

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 21, $C_2^3$

Order 6: $C_6$ x 21

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 576: $C_6.(S_3\times C_4^2)$ ($C_1$)

Order 288: $C_2^3.S_3^2$ ($C_2$) x 2, $C_6^2.C_2^3$ ($C_2$), $C_6^2.D_4$ ($C_2$), $C_6^2.D_4$ ($C_2$), $C_6^2.D_4$ ($C_2$), $C_6^2.D_4$ ($C_2$)

Order 144: $C_6^2.C_2^2$ ($C_4$) x 4, $C_6.D_{12}$ ($C_4$) x 4, $C_3^2:C_4^2$ ($C_4$) x 4, $C_6^2:C_4$ ($C_2^2$) x 4, $C_6^2:C_4$ ($C_2^2$) x 2, $C_2\times C_6\times C_{12}$ ($C_2^2$)

Order 96: $C_6.C_4^2$ ($S_3$), $C_2^3.D_6$ ($S_3$)

Order 72: $C_6:C_{12}$ ($C_2\times C_4$) x 12, $C_6.D_6$ ($C_2\times C_4$) x 4, $C_6\times C_{12}$ ($C_2\times C_4$) x 2, $C_6.D_6$ ($Q_8$) x 2, $C_6.D_6$ ($D_4$) x 2, $C_2\times C_6^2$ ($C_2^3$)

Order 48: $C_6.C_2^3$ ($D_6$) x 4, $C_6.D_4$ ($C_3:C_4$) x 4, $C_2^2\times C_{12}$ ($D_6$) x 2

Order 36: $C_3^2:C_4$ ($C_4:C_4$) x 4, $C_3:C_{12}$ ($C_4^2$) x 4, $C_6^2$ ($C_2^2\times C_4$) x 3, $C_6^2$ ($D_4:C_2$) x 2, $C_6^2$ ($C_2\times Q_8$), $C_6^2$ ($C_2\times D_4$)

Order 24: $C_6:C_4$ ($C_4\times S_3$) x 8, $C_6:C_4$ ($C_6:C_4$) x 4, $C_2\times C_{12}$ ($C_6:C_4$) x 2, $C_2\times C_{12}$ ($C_4\times S_3$) x 2, $C_2^2\times C_6$ ($C_2\times D_6$) x 2

Order 18: $C_3\times C_6$ ($C_4\times Q_8$) x 2, $C_3\times C_6$ ($C_4\times D_4$) x 2, $C_3\times C_6$ ($C_4^2:C_2$), $C_3\times C_6$ ($C_2^2.D_4$), $C_3\times C_6$ ($C_2\times C_4^2$)

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

Order 12: $C_2\times C_6$ ($C_4\times D_6$) x 5, $C_3:C_4$ ($C_{12}:C_4$) x 4, $C_2\times C_6$ ($D_4:S_3$) x 3, $C_2\times C_6$ ($S_3\times Q_8$) x 2, $C_2\times C_6$ ($S_3\times D_4$) x 2, $C_2\times C_6$ ($C_6.C_2^3$), $C_2\times C_6$ ($D_{12}:C_2$)

Order 9: $C_3^2$ ($C_4:C_4^2$)

Order 8: $C_2\times C_4$ ($C_6.D_6$) x 2, $C_2^3$ ($S_3\times D_6$)

Order 6: $C_6$ ($C_3:(C_4\times Q_8)$) x 3, $C_6$ ($D_6.D_4$) x 2, $C_6$ ($C_2^3.D_6$) x 2, $C_6$ ($(C_4\times S_3):C_4$), $C_6$ ($C_2^3.D_6$), $C_6$ ($S_3\times C_4^2$), $C_6$ ($C_3:C_4\times Q_8$), $C_6$ ($C_2^3.D_6$), $C_6$ ($C_6:C_4^2$), $C_6$ ($D_{12}:C_4$)

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

Order 3: $C_3$ ($C_{12}:C_4^2$), $C_3$ ($C_6.(C_4\times D_4)$)

Order 2: $C_2$ ($C_2^3.S_3^2$), $C_2$ ($C_6^2.C_2^3$), $C_2$ ($C_6^2.C_2^3$), $C_2$ ($C_6^2.C_2^3$), $C_2$ ($C_6^2.C_2^3$), $C_2$ ($C_6^2.C_2^3$), $C_2$ ($C_6^2.C_2^3$)

Order 1: $C_1$ ($C_6.(S_3\times C_4^2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 576: $C_6.(S_3\times C_4^2)$ ($C_1$)

Order 288: $C_2^3.S_3^2$ ($C_2$) x 2, $C_6^2.C_2^3$ ($C_2$), $C_6^2.D_4$ ($C_2$), $C_6^2.D_4$ ($C_2$), $C_6^2.D_4$ ($C_2$), $C_6^2.D_4$ ($C_2$)

Order 144: $C_6^2:C_4$ ($C_2^2$) x 4, $C_6^2:C_4$ ($C_2^2$) x 2, $C_6^2.C_2^2$ ($C_4$), $C_6.D_{12}$ ($C_4$), $C_3^2:C_4^2$ ($C_4$), $C_2\times C_6\times C_{12}$ ($C_2^2$)

Order 96: $C_6.C_4^2$ ($S_3$), $C_2^3.D_6$ ($S_3$)

Order 72: $C_6:C_{12}$ ($C_2\times C_4$) x 6, $C_6.D_6$ ($C_2\times C_4$) x 2, $C_2\times C_6^2$ ($C_2^3$), $C_6\times C_{12}$ ($C_2\times C_4$), $C_6.D_6$ ($Q_8$), $C_6.D_6$ ($D_4$)

Order 48: $C_6.C_2^3$ ($D_6$) x 4, $C_2^2\times C_{12}$ ($D_6$) x 2, $C_6.D_4$ ($C_3:C_4$)

Order 36: $C_6^2$ ($C_2^2\times C_4$) x 3, $C_6^2$ ($D_4:C_2$) x 2, $C_3^2:C_4$ ($C_4:C_4$), $C_3:C_{12}$ ($C_4^2$), $C_6^2$ ($C_2\times Q_8$), $C_6^2$ ($C_2\times D_4$)

Order 24: $C_6:C_4$ ($C_4\times S_3$) x 4, $C_6:C_4$ ($C_6:C_4$) x 2, $C_2^2\times C_6$ ($C_2\times D_6$) x 2, $C_2\times C_{12}$ ($C_6:C_4$), $C_2\times C_{12}$ ($C_4\times S_3$)

Order 18: $C_3\times C_6$ ($C_4\times Q_8$) x 2, $C_3\times C_6$ ($C_4\times D_4$) x 2, $C_3\times C_6$ ($C_4^2:C_2$), $C_3\times C_6$ ($C_2^2.D_4$), $C_3\times C_6$ ($C_2\times C_4^2$)

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

Order 12: $C_2\times C_6$ ($C_4\times D_6$) x 5, $C_2\times C_6$ ($D_4:S_3$) x 3, $C_2\times C_6$ ($S_3\times Q_8$) x 2, $C_2\times C_6$ ($S_3\times D_4$) x 2, $C_2\times C_6$ ($C_6.C_2^3$), $C_2\times C_6$ ($D_{12}:C_2$), $C_3:C_4$ ($C_{12}:C_4$)

Order 9: $C_3^2$ ($C_4:C_4^2$)

Order 8: $C_2^3$ ($S_3\times D_6$), $C_2\times C_4$ ($C_6.D_6$)

Order 6: $C_6$ ($C_3:(C_4\times Q_8)$) x 3, $C_6$ ($D_6.D_4$) x 2, $C_6$ ($C_2^3.D_6$) x 2, $C_6$ ($(C_4\times S_3):C_4$), $C_6$ ($C_2^3.D_6$), $C_6$ ($S_3\times C_4^2$), $C_6$ ($C_3:C_4\times Q_8$), $C_6$ ($C_2^3.D_6$), $C_6$ ($C_6:C_4^2$), $C_6$ ($D_{12}:C_4$)

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

Order 3: $C_3$ ($C_{12}:C_4^2$), $C_3$ ($C_6.(C_4\times D_4)$)

Order 2: $C_2$ ($C_2^3.S_3^2$), $C_2$ ($C_6^2.C_2^3$), $C_2$ ($C_6^2.C_2^3$), $C_2$ ($C_6^2.C_2^3$), $C_2$ ($C_6^2.C_2^3$), $C_2$ ($C_6^2.C_2^3$), $C_2$ ($C_6^2.C_2^3$)

Order 1: $C_1$ ($C_6.(S_3\times C_4^2)$)

Series

Derived series

$C_6.(S_3\times C_4^2)$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_1$

Chief series

$C_6.(S_3\times C_4^2)$

$\rhd$

$C_2^3.S_3^2$

$\rhd$

$C_6^2:C_4$

$\rhd$

$C_6:C_{12}$

$\rhd$

$C_3:C_{12}$

$\rhd$

$C_3:C_4$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6.(S_3\times C_4^2)$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_2^2\times C_4$

Supergroups

This group is a maximal subgroup of 12 larger groups in the database.

This group is a maximal quotient of 11 larger groups in the database.

Character theory

Complex character table

See the $96 \times 96$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $68 \times 68$ rational character table.