Abstract group 576.5055: $C_3:C_8\times S_4$

• Label: 576.5055
• Order: \( 2^{6} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $15$
• Trans deg.: $72$
• Rank: $2$

Group information

Description:	$C_3:C_8\times S_4$
Order:	\(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^3:D_6^2$, of order \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
Composition factors:	$C_2$ x 6, $C_3$ x 2
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	19	26	44	62	192	136	96	576
Conjugacy classes	1	5	3	10	7	16	14	4	60
Divisions	1	5	3	6	7	4	8	1	35
Autjugacy classes	1	4	3	5	6	4	7	1	31

Dimension	1	2	3	4	6	8	12
Irr. complex chars.	16	16	16	4	8	0	0	60
Irr. rational chars.	4	8	4	7	6	2	4	35

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$72$
Rank:	$2$
Inequivalent generating pairs:	$36$

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:	$\langle a, b, c, d \mid a^{2}=b^{24}=c^{2}=d^{6}=[a,c]=[c,d]=1, b^{a}=b^{17}, d^{a}=cd, c^{b}=d^{3}, d^{b}=cd^{5} \rangle$
Permutation group:	Degree $15$ $\langle(5,6,8,10,7,9,11,12)(14,15), (2,3)(5,7)(6,9)(8,11)(10,12), (5,8,7,11)(6,10,9,12) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 5 & 4 \\ 12 & 5 \end{array}\right), \left(\begin{array}{rr} 11 & 0 \\ 10 & 11 \end{array}\right), \left(\begin{array}{rr} 13 & 4 \\ 8 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 10 \\ 10 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 6 & 5 \\ 5 & 16 \end{array}\right), \left(\begin{array}{rr} 11 & 15 \\ 5 & 16 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Direct product:	$(C_3:C_8)$ $\, \times\, $ $S_4$
Semidirect product:	$(C_3\times S_4)$ $\,\rtimes\,$ $C_8$ (2)	$C_3$ $\,\rtimes\,$ $(C_8\times S_4)$	$A_4$ $\,\rtimes\,$ $(C_6:C_8)$	$(C_{12}.S_4)$ $\,\rtimes\,$ $C_2$	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_4\times S_4)$ . $S_3$	$C_4$ . $(S_3\times S_4)$	$(C_6\times S_4)$ . $C_4$	$C_6$ . $(C_4\times S_4)$	all 18

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

Homology

Abelianization:	$C_{2} \times C_{8} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 700 subgroups in 166 conjugacy classes, 33 normal (29 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $S_3\times S_4$
Commutator:	$G' \simeq$ $C_3\times A_4$	$G/G' \simeq$ $C_2\times C_8$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $S_3\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{12}$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_3:C_8\times S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_4\times S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8\times D_4$
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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 576: $C_3:C_8\times S_4$

Order 288: $C_{12}\times S_4$, $A_4\times C_3:C_8$, $C_{12}.S_4$

Order 192: $C_8\times S_4$, $C_4^2.D_6$

Order 144: $C_{12}.D_6$, $C_6\times S_4$, $C_{12}\times A_4$, $A_4:C_{12}$

Order 96: $A_4:C_8$ x 2, $C_{12}.D_4$ x 2, $C_{12}.C_2^3$ x 2, $C_{12}:C_8$, $C_8\times A_4$, $C_4\times S_4$, $D_4\times C_{12}$, $C_{12}:C_8$

Order 72: $C_3\times S_4$ x 2, $C_6\times A_4$, $S_3\times C_{12}$, $C_3^2:C_8$, $C_3:C_{24}$

Order 64: $C_8\times D_4$

Order 48: $C_6:C_8$ x 6, $C_2^2\times C_{12}$ x 2, $C_4\times A_4$ x 2, $C_2^2:C_{12}$ x 2, $C_2\times S_4$, $C_6\times D_4$, $S_3\times C_8$, $A_4:C_4$, $C_4:C_{12}$, $C_4\times C_{12}$

Order 36: $C_3\times C_{12}$, $C_3:C_{12}$, $C_6\times S_3$, $C_3\times A_4$

Order 32: $C_2^2:C_8$ x 2, $C_2^2\times C_8$ x 2, $C_4\times C_8$, $C_4\times D_4$, $C_4:C_8$

Order 24: $C_2\times C_{12}$ x 7, $C_3:C_8$ x 6, $C_3\times D_4$ x 4, $C_2^2\times C_6$ x 2, $C_2\times A_4$ x 2, $S_4$ x 2, $C_4\times S_3$, $C_{24}$

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

Order 16: $C_2\times C_8$ x 6, $C_2^2:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_4:C_4$, $C_4^2$, $C_2\times D_4$

Order 12: $C_{12}$ x 8, $C_2\times C_6$ x 7, $A_4$ x 2, $D_6$, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 7, $D_4$ x 4, $C_8$ x 4, $C_2^3$ x 2

Order 6: $C_6$ x 7, $S_3$ x 2

Order 4: $C_2^2$ x 7, $C_4$ x 6

Order 3: $C_3$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 576: $C_3:C_8\times S_4$

Order 288: $C_{12}\times S_4$, $A_4\times C_3:C_8$, $C_{12}.S_4$

Order 192: $C_8\times S_4$, $C_4^2.D_6$

Order 144: $C_{12}.D_6$, $C_6\times S_4$, $C_{12}\times A_4$, $A_4:C_{12}$

Order 96: $A_4:C_8$ x 2, $C_{12}.D_4$ x 2, $C_{12}.C_2^3$ x 2, $C_{12}:C_8$, $C_8\times A_4$, $C_4\times S_4$, $D_4\times C_{12}$, $C_{12}:C_8$

Order 72: $C_6\times A_4$, $C_3\times S_4$, $S_3\times C_{12}$, $C_3^2:C_8$, $C_3:C_{24}$

Order 64: $C_8\times D_4$

Order 48: $C_6:C_8$ x 6, $C_2^2\times C_{12}$ x 2, $C_4\times A_4$ x 2, $C_2^2:C_{12}$ x 2, $C_2\times S_4$, $C_6\times D_4$, $S_3\times C_8$, $A_4:C_4$, $C_4:C_{12}$, $C_4\times C_{12}$

Order 36: $C_3\times C_{12}$, $C_3:C_{12}$, $C_6\times S_3$, $C_3\times A_4$

Order 32: $C_2^2:C_8$ x 2, $C_2^2\times C_8$ x 2, $C_4\times C_8$, $C_4\times D_4$, $C_4:C_8$

Order 24: $C_2\times C_{12}$ x 7, $C_3:C_8$ x 6, $C_2^2\times C_6$ x 2, $C_2\times A_4$ x 2, $C_3\times D_4$ x 2, $C_4\times S_3$, $C_{24}$, $S_4$

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

Order 16: $C_2\times C_8$ x 6, $C_2^2:C_4$ x 2, $C_2^2\times C_4$ x 2, $C_4:C_4$, $C_4^2$, $C_2\times D_4$

Order 12: $C_{12}$ x 7, $C_2\times C_6$ x 6, $A_4$ x 2, $D_6$, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 7, $C_8$ x 4, $C_2^3$ x 2, $D_4$ x 2

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

Order 4: $C_2^2$ x 6, $C_4$ x 5

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 576: $C_3:C_8\times S_4$ ($C_1$)

Order 288: $C_{12}\times S_4$ ($C_2$), $A_4\times C_3:C_8$ ($C_2$), $C_{12}.S_4$ ($C_2$)

Order 144: $C_6\times S_4$ ($C_4$), $C_{12}\times A_4$ ($C_2^2$), $A_4:C_{12}$ ($C_4$)

Order 96: $C_4\times S_4$ ($S_3$), $C_{12}.C_2^3$ ($S_3$)

Order 72: $C_3\times S_4$ ($C_8$) x 2, $C_6\times A_4$ ($C_2\times C_4$)

Order 48: $C_2\times S_4$ ($C_3:C_4$), $C_2^2\times C_{12}$ ($D_6$), $C_4\times A_4$ ($D_6$), $A_4:C_4$ ($C_3:C_4$)

Order 36: $C_3\times A_4$ ($C_2\times C_8$)

Order 24: $S_4$ ($C_3:C_8$) x 2, $C_2^2\times C_6$ ($C_4\times S_3$), $C_2\times A_4$ ($C_6:C_4$), $C_3:C_8$ ($S_4$)

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

Order 12: $C_2\times C_6$ ($S_3\times C_8$), $A_4$ ($C_6:C_8$), $C_{12}$ ($C_2\times S_4$)

Order 8: $C_2^3$ ($C_6.D_6$)

Order 6: $C_6$ ($C_4\times S_4$)

Order 4: $C_2^2$ ($C_{12}.D_6$), $C_4$ ($S_3\times S_4$)

Order 3: $C_3$ ($C_8\times S_4$)

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

Order 1: $C_1$ ($C_3:C_8\times S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 576: $C_3:C_8\times S_4$ ($C_1$)

Order 288: $C_{12}\times S_4$ ($C_2$), $A_4\times C_3:C_8$ ($C_2$), $C_{12}.S_4$ ($C_2$)

Order 144: $C_6\times S_4$ ($C_4$), $C_{12}\times A_4$ ($C_2^2$), $A_4:C_{12}$ ($C_4$)

Order 96: $C_4\times S_4$ ($S_3$), $C_{12}.C_2^3$ ($S_3$)

Order 72: $C_6\times A_4$ ($C_2\times C_4$), $C_3\times S_4$ ($C_8$)

Order 48: $C_2\times S_4$ ($C_3:C_4$), $C_2^2\times C_{12}$ ($D_6$), $C_4\times A_4$ ($D_6$), $A_4:C_4$ ($C_3:C_4$)

Order 36: $C_3\times A_4$ ($C_2\times C_8$)

Order 24: $C_2^2\times C_6$ ($C_4\times S_3$), $C_2\times A_4$ ($C_6:C_4$), $S_4$ ($C_3:C_8$), $C_3:C_8$ ($S_4$)

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

Order 12: $C_2\times C_6$ ($S_3\times C_8$), $A_4$ ($C_6:C_8$), $C_{12}$ ($C_2\times S_4$)

Order 8: $C_2^3$ ($C_6.D_6$)

Order 6: $C_6$ ($C_4\times S_4$)

Order 4: $C_2^2$ ($C_{12}.D_6$), $C_4$ ($S_3\times S_4$)

Order 3: $C_3$ ($C_8\times S_4$)

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

Order 1: $C_1$ ($C_3:C_8\times S_4$)

Series

Derived series

$C_3:C_8\times S_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_3:C_8\times S_4$

$\rhd$

$C_{12}\times S_4$

$\rhd$

$C_{12}\times A_4$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3:C_8\times S_4$

$\rhd$

$C_3\times A_4$

Upper central series

$C_1$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 14 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 $60 \times 60$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $35 \times 35$ rational character table.