Abstract group 144.189: $C_6:S_4$

• Label: 144.189
• Order: \( 2^{4} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $9$
• Trans deg.: $18$
• Rank: $3$

Group information

Description:	$C_6:S_4$
Order:	\(144\)\(\medspace = 2^{4} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2\times C_6^2:D_6$, of order \(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Composition factors:	$C_2$ x 4, $C_3$ x 2
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6
Elements	1	43	26	36	38	144
Conjugacy classes	1	5	4	2	6	18
Divisions	1	5	4	2	6	18
Autjugacy classes	1	4	2	1	4	12

Dimension	1	2	3	6
Irr. complex chars.	4	8	4	2	18
Irr. rational chars.	4	8	4	2	18

Minimal presentations

Permutation degree:	$9$
Transitive degree:	$18$
Rank:	$3$
Inequivalent generating triples:	$1260$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	6	6
Arbitrary	5	5	5

Constructions

Presentation:	$\langle a, b, c, d \mid a^{2}=b^{6}=c^{2}=d^{6}=[a,c]=[c,d]=1, b^{a}=b^{5}, d^{a}=cd^{5}, c^{b}=cd^{3}, d^{b}=cd^{4} \rangle$
Permutation group:	$\langle(2,3)(6,7)(8,9), (8,9), (2,3,4)(5,6,7), (5,6,7), (1,2)(3,4), (1,3)(2,4)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & -1 & -1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & -1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & -1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & -1 \end{array}\right) \right\rangle \subseteq \GL_{5}(\Z)$
	$\left\langle \left(\begin{array}{rr} 7 & 0 \\ 6 & 7 \end{array}\right), \left(\begin{array}{rr} 1 & 9 \\ 9 & 10 \end{array}\right), \left(\begin{array}{rr} 1 & 5 \\ 6 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 6 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/12\Z)$
Transitive group:	18T66	24T251	36T109	36T133	all 6
Direct product:	$C_2$ $\, \times\, $ $(C_3:S_4)$
Semidirect product:	$C_6$ $\,\rtimes\,$ $S_4$	$A_4$ $\,\rtimes\,$ $D_6$ (3)	$(C_2\times A_4)$ $\,\rtimes\,$ $S_3$ (3)	$C_3$ $\,\rtimes\,$ $(C_2\times S_4)$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 9.

Homology

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

Subgroups

There are 450 subgroups in 86 conjugacy classes, 19 normal (11 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 144: $C_6:S_4$

Order 72: $C_3:S_4$ x 2, $C_6\times A_4$

Order 48: $C_2\times S_4$ x 3, $C_6:D_4$

Order 36: $C_6:S_3$, $C_3\times A_4$

Order 24: $S_4$ x 6, $C_3:D_4$ x 4, $C_2\times A_4$ x 3, $C_6:C_4$, $C_2^2\times C_6$, $C_2\times D_6$

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

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 7, $C_2\times C_6$ x 3, $A_4$ x 3, $C_3:C_4$ x 2

Order 9: $C_3^2$

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

Order 6: $S_3$ x 8, $C_6$ x 6

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

Order 3: $C_3$ x 4

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 144: $C_6:S_4$

Order 72: $C_6\times A_4$, $C_3:S_4$

Order 48: $C_2\times S_4$, $C_6:D_4$

Order 36: $C_6:S_3$, $C_3\times A_4$

Order 24: $C_3:D_4$ x 2, $C_6:C_4$, $C_2^2\times C_6$, $C_2\times D_6$, $C_2\times A_4$, $S_4$

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

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 4, $C_2\times C_6$ x 3, $A_4$, $C_3:C_4$

Order 9: $C_3^2$

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

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

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 144: $C_6:S_4$ ($C_1$)

Order 72: $C_3:S_4$ ($C_2$) x 2, $C_6\times A_4$ ($C_2$)

Order 36: $C_3\times A_4$ ($C_2^2$)

Order 24: $C_2\times A_4$ ($S_3$) x 3, $C_2^2\times C_6$ ($S_3$)

Order 12: $A_4$ ($D_6$) x 3, $C_2\times C_6$ ($D_6$)

Order 8: $C_2^3$ ($C_3:S_3$)

Order 6: $C_6$ ($S_4$)

Order 4: $C_2^2$ ($C_6:S_3$)

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

Order 2: $C_2$ ($C_3:S_4$)

Order 1: $C_1$ ($C_6:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 144: $C_6:S_4$ ($C_1$)

Order 72: $C_6\times A_4$ ($C_2$), $C_3:S_4$ ($C_2$)

Order 36: $C_3\times A_4$ ($C_2^2$)

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

Order 12: $C_2\times C_6$ ($D_6$), $A_4$ ($D_6$)

Order 8: $C_2^3$ ($C_3:S_3$)

Order 6: $C_6$ ($S_4$)

Order 4: $C_2^2$ ($C_6:S_3$)

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

Order 2: $C_2$ ($C_3:S_4$)

Order 1: $C_1$ ($C_6:S_4$)

Series

Derived series

$C_6:S_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_6:S_4$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_6:S_4$

$\rhd$

$C_3\times A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

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

This group is a maximal quotient of 52 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $18 \times 18$ rational character table. Alternatively, you may search for characters of this group with desired properties.