Abstract group 288.402: $C_{12}.S_4$

• Label: 288.402
• Order: \( 2^{5} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $72$
• Rank: $3$

Group information

Description:	$C_{12}.S_4$
Order:	\(288\)\(\medspace = 2^{5} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^2\times C_6^2:D_6$, of order \(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Composition factors:	$C_2$ x 5, $C_3$ x 2
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	7	26	8	38	144	64	288
Conjugacy classes	1	3	4	4	6	8	10	36
Divisions	1	3	4	2	6	2	5	23
Autjugacy classes	1	3	2	2	4	2	3	17

Dimension	1	2	3	4	6	12
Irr. complex chars.	8	16	8	0	4	0	36
Irr. rational chars.	2	9	2	5	3	2	23

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$72$
Rank:	$3$
Inequivalent generating triples:	$6720$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	12
Arbitrary	5	7	7

Constructions

Presentation:	$\langle a, b, c, d \mid a^{8}=b^{3}=c^{6}=d^{2}=[a,d]=[c,d]=1, b^{a}=b^{2}, c^{a}=c^{5}d, c^{b}=c^{4}d, d^{b}=c^{3}d \rangle$
Permutation group:	Degree $15$ $\langle(1,2,3,5,4,6,7,8)(10,11)(13,14), (1,3,4,7)(2,5,6,8), (1,4)(2,6)(3,7)(5,8), (9,10,11)(13,14,15), (9,11,10), (12,13)(14,15), (12,14)(13,15)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 13 & 4 \\ 8 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 10 & 17 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 7 & 10 \\ 10 & 7 \end{array}\right), \left(\begin{array}{rr} 11 & 15 \\ 5 & 16 \end{array}\right), \left(\begin{array}{rr} 0 & 7 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3\times A_4)$ $\,\rtimes\,$ $C_8$	$A_4$ $\,\rtimes\,$ $(C_3:C_8)$ (3)	$C_3$ $\,\rtimes\,$ $(A_4:C_8)$	$C_2^2$ $\,\rtimes\,$ $(C_3^2:C_8)$	all 5
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{12}$ . $S_4$	$(C_4\times A_4)$ . $S_3$ (3)	$C_4$ . $(C_3:S_4)$	$C_2$ . $(C_6.S_4)$	all 12

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

Homology

Abelianization:	$C_{8} $
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 248 subgroups in 68 conjugacy classes, 25 normal (16 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $C_3:S_4$
Commutator:	$G' \simeq$ $C_3\times A_4$	$G/G' \simeq$ $C_8$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_3:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{12}$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_{12}.S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_3:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:C_8$
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 288: $C_{12}.S_4$

Order 144: $C_{12}\times A_4$

Order 96: $A_4:C_8$ x 3, $C_{12}.D_4$

Order 72: $C_6\times A_4$, $C_3^2:C_8$

Order 48: $C_4\times A_4$ x 3, $C_6:C_8$ x 2, $C_2^2\times C_{12}$

Order 36: $C_3\times C_{12}$, $C_3\times A_4$

Order 32: $C_2^2:C_8$

Order 24: $C_3:C_8$ x 5, $C_2\times A_4$ x 3, $C_2\times C_{12}$ x 2, $C_2^2\times C_6$

Order 18: $C_3\times C_6$

Order 16: $C_2\times C_8$ x 2, $C_2^2\times C_4$

Order 12: $C_{12}$ x 5, $C_2\times C_6$ x 3, $A_4$ x 3

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 2, $C_8$ x 2, $C_2^3$

Order 6: $C_6$ x 6

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

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 288: $C_{12}.S_4$

Order 144: $C_{12}\times A_4$

Order 96: $A_4:C_8$, $C_{12}.D_4$

Order 72: $C_6\times A_4$, $C_3^2:C_8$

Order 48: $C_6:C_8$ x 2, $C_2^2\times C_{12}$, $C_4\times A_4$

Order 36: $C_3\times C_{12}$, $C_3\times A_4$

Order 32: $C_2^2:C_8$

Order 24: $C_3:C_8$ x 3, $C_2\times C_{12}$ x 2, $C_2^2\times C_6$, $C_2\times A_4$

Order 18: $C_3\times C_6$

Order 16: $C_2\times C_8$ x 2, $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 3, $C_{12}$ x 3, $A_4$

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 2, $C_8$ x 2, $C_2^3$

Order 6: $C_6$ x 4

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 288: $C_{12}.S_4$ ($C_1$)

Order 144: $C_{12}\times A_4$ ($C_2$)

Order 72: $C_6\times A_4$ ($C_4$)

Order 48: $C_4\times A_4$ ($S_3$) x 3, $C_2^2\times C_{12}$ ($S_3$)

Order 36: $C_3\times A_4$ ($C_8$)

Order 24: $C_2\times A_4$ ($C_3:C_4$) x 3, $C_2^2\times C_6$ ($C_3:C_4$)

Order 16: $C_2^2\times C_4$ ($C_3:S_3$)

Order 12: $A_4$ ($C_3:C_8$) x 3, $C_2\times C_6$ ($C_3:C_8$), $C_{12}$ ($S_4$)

Order 8: $C_2^3$ ($C_3^2:C_4$)

Order 6: $C_6$ ($A_4:C_4$)

Order 4: $C_2^2$ ($C_3^2:C_8$), $C_4$ ($C_3:S_4$)

Order 3: $C_3$ ($A_4:C_8$)

Order 2: $C_2$ ($C_6.S_4$)

Order 1: $C_1$ ($C_{12}.S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 288: $C_{12}.S_4$ ($C_1$)

Order 144: $C_{12}\times A_4$ ($C_2$)

Order 72: $C_6\times A_4$ ($C_4$)

Order 48: $C_2^2\times C_{12}$ ($S_3$), $C_4\times A_4$ ($S_3$)

Order 36: $C_3\times A_4$ ($C_8$)

Order 24: $C_2^2\times C_6$ ($C_3:C_4$), $C_2\times A_4$ ($C_3:C_4$)

Order 16: $C_2^2\times C_4$ ($C_3:S_3$)

Order 12: $C_2\times C_6$ ($C_3:C_8$), $A_4$ ($C_3:C_8$), $C_{12}$ ($S_4$)

Order 8: $C_2^3$ ($C_3^2:C_4$)

Order 6: $C_6$ ($A_4:C_4$)

Order 4: $C_2^2$ ($C_3^2:C_8$), $C_4$ ($C_3:S_4$)

Order 3: $C_3$ ($A_4:C_8$)

Order 2: $C_2$ ($C_6.S_4$)

Order 1: $C_1$ ($C_{12}.S_4$)

Series

Derived series

$C_{12}.S_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_{12}.S_4$

$\rhd$

$C_{12}\times A_4$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_{12}.S_4$

$\rhd$

$C_3\times A_4$

Upper central series

$C_1$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 25 larger groups in the database.

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

Character theory

Complex character table

See the $36 \times 36$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $23 \times 23$ rational character table.