Abstract group 864.4010: $C_6^2.S_4$

• Label: 864.4010
• Order: \( 2^{5} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $17$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_6^2.S_4$
Order:	\(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^2.S_4\times S_4$, of order \(5184\)\(\medspace = 2^{6} \cdot 3^{4} \)
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	9	12	18
Elements	1	87	8	72	264	72	144	216	864
Conjugacy classes	1	11	3	4	37	3	8	9	76
Divisions	1	11	2	4	22	2	4	6	52
Autjugacy classes	1	4	3	1	15	3	2	3	32

Dimension	1	2	3	4	6	12
Irr. complex chars.	24	12	24	0	16	0	76
Irr. rational chars.	8	12	8	4	16	4	52

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$32760$

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:	$\langle a, b, c, d, e \mid a^{2}=b^{6}=c^{18}=d^{2}=e^{2}=[a,b]=[a,c]=[a,d]=[a,e]=[d,e]=1, c^{b}=c^{11}, d^{b}=e, e^{b}=d, d^{c}=de, e^{c}=d \rangle$
Permutation group:	Degree $17$ $\langle(2,3)(5,6)(7,8)(10,12)(11,15)(13,14)(16,17), (5,6)(7,8), (5,7)(6,8), (10,13,17) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 18 & 19 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 7 & 13 \\ 6 & 11 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 18 & 1 \end{array}\right), \left(\begin{array}{rr} 7 & 13 \\ 3 & 28 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/36\Z)$
Transitive group:	36T1297				more information
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(C_3^2.S_4)$
Semidirect product:	$(C_6^2.A_4)$ $\,\rtimes\,$ $C_2$	$C_2^4$ $\,\rtimes\,$ $(C_9:C_6)$	$(C_2^3:D_{18})$ $\,\rtimes\,$ $C_3$	$(C_2^2:D_{18})$ $\,\rtimes\,$ $C_6$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $S_4$	$C_6$ . $(C_6\times S_4)$	$(C_2\times C_6^2)$ . $D_6$	$C_6^2$ . $(C_2\times D_6)$	all 12
Aut. group:	$\Aut(C_{12}.S_4)$	$\Aut(C_6.\GL(2,3))$	$\Aut(C_{12}.S_4)$	$\Aut(C_{12}.S_4)$	all 18

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{36}\Z)$.

Homology

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

Subgroups

There are 2796 subgroups in 489 conjugacy classes, 62 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_3^2.S_4$
Commutator:	$G' \simeq$ $C_2^2:C_9$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_2\times C_6\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_6^2$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_6^2.S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times C_6$	$G/\operatorname{soc} \simeq$ $C_3\times S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9:C_3$

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: $C_6^2.S_4$

Order 432: $C_6^2.D_6$ x 6, $C_6^2.A_4$

Order 288: $C_2^3:D_{18}$, $C_6^2:D_4$

Order 216: $C_3^2.S_4$ x 4, $C_6^2.C_6$ x 3, $D_{18}:C_6$

Order 144: $C_6^2:C_2^2$ x 12, $C_2^2:D_{18}$ x 6, $C_2^3:C_{18}$ x 2, $C_2^2\times C_6^2$, $C_6^2:C_2^2$, $C_6^2:C_4$

Order 108: $C_{18}:C_6$ x 6, $C_{18}:C_6$, $C_3^2.A_4$

Order 96: $C_{12}:C_2^3$, $C_2^3:D_6$

Order 72: $C_6\wr C_2$ x 16, $C_6\times D_6$ x 10, $C_2\times C_6^2$ x 7, $C_6:C_{12}$ x 6, $C_2^2:C_{18}$ x 6, $C_2^2:D_9$ x 4, $C_2\times D_{18}$

Order 54: $C_9:C_6$ x 4, $C_9:C_6$ x 3

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

Order 36: $C_6\times S_3$ x 16, $C_6^2$ x 12, $D_{18}$ x 6, $C_3:C_{12}$ x 4, $C_2\times C_{18}$ x 2, $C_2^2:C_9$ x 2

Order 32: $C_2^2\times D_4$

Order 27: $C_9:C_3$

Order 24: $C_2^2\times C_6$ x 28, $C_3:D_4$ x 16, $C_3\times D_4$ x 16, $C_2\times D_6$ x 10, $C_2\times C_{12}$ x 6, $C_6:C_4$ x 6

Order 18: $C_3\times C_6$ x 7, $C_{18}$ x 6, $C_3\times S_3$ x 4, $D_9$ x 4

Order 16: $C_2\times D_4$ x 12, $C_2^4$ x 2, $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 51, $D_6$ x 16, $C_{12}$ x 4, $C_3:C_4$ x 4

Order 9: $C_9$ x 2, $C_3^2$

Order 8: $C_2^3$ x 17, $D_4$ x 16, $C_2\times C_4$ x 6

Order 6: $C_6$ x 22, $S_3$ x 4

Order 4: $C_2^2$ x 28, $C_4$ x 4

Order 3: $C_3$ x 2

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 864: $C_6^2.S_4$

Order 432: $C_6^2.A_4$, $C_6^2.D_6$

Order 288: $C_2^3:D_{18}$, $C_6^2:D_4$

Order 216: $C_3^2.S_4$, $D_{18}:C_6$, $C_6^2.C_6$

Order 144: $C_6^2:C_2^2$ x 2, $C_2^3:C_{18}$ x 2, $C_2^2\times C_6^2$, $C_6^2:C_2^2$, $C_6^2:C_4$, $C_2^2:D_{18}$

Order 108: $C_{18}:C_6$, $C_{18}:C_6$, $C_3^2.A_4$

Order 96: $C_{12}:C_2^3$, $C_2^3:D_6$

Order 72: $C_2\times C_6^2$ x 3, $C_6\times D_6$ x 3, $C_6\wr C_2$ x 2, $C_2^2:C_{18}$ x 2, $C_6:C_{12}$, $C_2\times D_{18}$, $C_2^2:D_9$

Order 54: $C_9:C_6$, $C_9:C_6$

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

Order 36: $C_6^2$ x 6, $C_6\times S_3$ x 3, $C_2\times C_{18}$ x 2, $C_2^2:C_9$ x 2, $C_3:C_{12}$, $D_{18}$

Order 32: $C_2^2\times D_4$

Order 27: $C_9:C_3$

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

Order 18: $C_3\times C_6$ x 3, $C_{18}$ x 2, $C_3\times S_3$, $D_9$

Order 16: $C_2^4$ x 2, $C_2\times D_4$ x 2, $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 18, $D_6$ x 3, $C_{12}$, $C_3:C_4$

Order 9: $C_9$ x 2, $C_3^2$

Order 8: $C_2^3$ x 6, $D_4$ x 2, $C_2\times C_4$

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

Order 4: $C_2^2$ x 9, $C_4$

Order 3: $C_3$ x 2

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 864: $C_6^2.S_4$ ($C_1$)

Order 432: $C_6^2.D_6$ ($C_2$) x 6, $C_6^2.A_4$ ($C_2$)

Order 288: $C_2^3:D_{18}$ ($C_3$)

Order 216: $C_3^2.S_4$ ($C_2^2$) x 4, $C_6^2.C_6$ ($C_2^2$) x 3

Order 144: $C_2^2:D_{18}$ ($C_6$) x 6, $C_2^2\times C_6^2$ ($S_3$), $C_2^3:C_{18}$ ($C_6$)

Order 108: $C_3^2.A_4$ ($C_2^3$)

Order 72: $C_2^2:D_9$ ($C_2\times C_6$) x 4, $C_2\times C_6^2$ ($D_6$) x 3, $C_2^2:C_{18}$ ($C_2\times C_6$) x 3

Order 48: $C_2^3\times C_6$ ($C_3\times S_3$)

Order 36: $C_2^2:C_9$ ($C_2^2\times C_6$), $C_6^2$ ($C_2\times D_6$), $C_6^2$ ($S_4$)

Order 24: $C_2^2\times C_6$ ($C_6\times S_3$) x 3

Order 18: $C_3\times C_6$ ($C_2\times S_4$) x 3

Order 16: $C_2^4$ ($C_9:C_6$)

Order 12: $C_2\times C_6$ ($C_6\times D_6$), $C_2\times C_6$ ($C_3\times S_4$)

Order 9: $C_3^2$ ($C_2^2\times S_4$)

Order 8: $C_2^3$ ($C_{18}:C_6$) x 3

Order 6: $C_6$ ($C_6\times S_4$) x 3

Order 4: $C_2^2$ ($C_3^2.S_4$), $C_2^2$ ($D_{18}:C_6$)

Order 3: $C_3$ ($C_2\times C_6\times S_4$)

Order 2: $C_2$ ($C_6^2.D_6$) x 3

Order 1: $C_1$ ($C_6^2.S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 864: $C_6^2.S_4$ ($C_1$)

Order 432: $C_6^2.A_4$ ($C_2$), $C_6^2.D_6$ ($C_2$)

Order 288: $C_2^3:D_{18}$ ($C_3$)

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

Order 144: $C_2^2\times C_6^2$ ($S_3$), $C_2^3:C_{18}$ ($C_6$), $C_2^2:D_{18}$ ($C_6$)

Order 108: $C_3^2.A_4$ ($C_2^3$)

Order 72: $C_2\times C_6^2$ ($D_6$), $C_2^2:C_{18}$ ($C_2\times C_6$), $C_2^2:D_9$ ($C_2\times C_6$)

Order 48: $C_2^3\times C_6$ ($C_3\times S_3$)

Order 36: $C_2^2:C_9$ ($C_2^2\times C_6$), $C_6^2$ ($C_2\times D_6$), $C_6^2$ ($S_4$)

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

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

Order 16: $C_2^4$ ($C_9:C_6$)

Order 12: $C_2\times C_6$ ($C_6\times D_6$), $C_2\times C_6$ ($C_3\times S_4$)

Order 9: $C_3^2$ ($C_2^2\times S_4$)

Order 8: $C_2^3$ ($C_{18}:C_6$)

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

Order 4: $C_2^2$ ($C_3^2.S_4$), $C_2^2$ ($D_{18}:C_6$)

Order 3: $C_3$ ($C_2\times C_6\times S_4$)

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

Order 1: $C_1$ ($C_6^2.S_4$)

Series

Derived series

$C_6^2.S_4$

$\rhd$

$C_2^2:C_9$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_6^2.S_4$

$\rhd$

$C_6^2.A_4$

$\rhd$

$C_6^2.C_6$

$\rhd$

$C_2^2:C_{18}$

$\rhd$

$C_2^2:C_9$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_6^2.S_4$

$\rhd$

$C_2^2:C_9$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 13 larger groups in the database.

This group is a maximal quotient of 28 larger groups in the database.

Character theory

Complex character table

See the $76 \times 76$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $52 \times 52$ rational character table.