Abstract group $C_7:C_3\times S_4$

• Label: 504.159
• Order: \( 2^{3} \cdot 3^{2} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{4} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $11$
• Trans deg.: $28$
• Rank: $2$

Group information

Description:	$C_7:C_3\times S_4$
Order:	\(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$S_4\times F_7$, of order \(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \) (generators)
Outer automorphisms:	$C_2$, of order \(2\)
Composition factors:	$C_2$ x 3, $C_3$ x 2, $C_7$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	7	12	14	21	28
Elements	1	9	134	6	126	6	84	54	48	36	504
Conjugacy classes	1	2	5	1	4	2	2	4	2	2	25
Divisions	1	2	3	1	2	1	1	2	1	1	15
Autjugacy classes	1	2	5	1	4	1	2	2	1	1	20

Dimension	1	2	3	4	6	9	12	18
Irr. complex chars.	6	3	10	0	2	4	0	0	25
Irr. rational chars.	2	3	2	1	4	0	1	2	15

Minimal Presentations

Permutation degree:	$11$
Transitive degree:	$28$
Rank:	$2$
Inequivalent generating pairs:	$72$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	9	18	18
Arbitrary	6	9	9

Constructions

Presentation:	$\langle a, b, c, d \mid a^{6}=b^{3}=c^{14}=d^{2}=[a,d]=[c,d]=1, b^{a}=b^{2}, c^{a}=c^{9}d, c^{b}=c^{8}d, d^{b}=c^{7}d \rangle$
Permutation group:	Degree $11$ $\langle(2,3), (6,7,8)(9,11,10), (2,3,4), (5,6,7,9,8,10,11), (1,2)(3,4), (1,3)(2,4)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 8 & 21 \\ 21 & 8 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 14 & 1 \end{array}\right), \left(\begin{array}{rr} 15 & 0 \\ 14 & 15 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 22 & 7 \\ 7 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 25 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/28\Z)$
Transitive group:	28T69	42T93	42T94		more information
Direct product:	$(C_7:C_3)$ $\, \times\, $ $S_4$
Semidirect product:	$(C_7\times S_4)$ $\,\rtimes\,$ $C_3$	$C_7$ $\,\rtimes\,$ $(C_3\times S_4)$	$(C_7\times A_4)$ $\,\rtimes\,$ $C_6$	$A_4$ $\,\rtimes\,$ $(C_7:C_6)$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

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

Subgroups

There are 380 subgroups in 48 conjugacy classes, 12 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_7:C_3\times S_4$
Commutator:	$G' \simeq$ $C_7\times A_4$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_7:C_3\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{14}$	$G/\operatorname{Fit} \simeq$ $C_3\times S_3$
Radical:	$R \simeq$ $C_7:C_3\times S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{14}$	$G/\operatorname{soc} \simeq$ $C_3\times S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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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Classes of subgroups up to conjugation

Order 504: $C_7:C_3\times S_4$

Order 252: $A_4\times C_7:C_3$

Order 168: $C_7\times S_4$, $C_{28}:C_6$

Order 126: $C_{21}:C_6$

Order 84: $C_{14}:C_6$ x 2, $C_7:C_{12}$, $C_7:A_4$, $C_7\times A_4$

Order 72: $C_3\times S_4$

Order 63: $C_{21}:C_3$

Order 56: $C_7\times D_4$

Order 42: $C_7:C_6$ x 2, $S_3\times C_7$

Order 36: $C_3\times A_4$

Order 28: $C_2\times C_{14}$ x 2, $C_{28}$

Order 24: $S_4$, $C_3\times D_4$

Order 21: $C_7:C_3$ x 2, $C_{21}$

Order 18: $C_3\times S_3$

Order 14: $C_{14}$ x 2

Order 12: $C_2\times C_6$ x 2, $A_4$ x 2, $C_{12}$

Order 9: $C_3^2$

Order 8: $D_4$

Order 7: $C_7$

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

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 504: $C_7:C_3\times S_4$

Order 252: $A_4\times C_7:C_3$

Order 168: $C_7\times S_4$, $C_{28}:C_6$

Order 126: $C_{21}:C_6$

Order 84: $C_{14}:C_6$ x 2, $C_7:C_{12}$, $C_7:A_4$, $C_7\times A_4$

Order 72: $C_3\times S_4$

Order 63: $C_{21}:C_3$

Order 56: $C_7\times D_4$

Order 42: $C_7:C_6$ x 2, $S_3\times C_7$

Order 36: $C_3\times A_4$

Order 28: $C_2\times C_{14}$ x 2, $C_{28}$

Order 24: $S_4$, $C_3\times D_4$

Order 21: $C_7:C_3$ x 2, $C_{21}$

Order 18: $C_3\times S_3$

Order 14: $C_{14}$ x 2

Order 12: $C_2\times C_6$ x 2, $A_4$ x 2, $C_{12}$

Order 9: $C_3^2$

Order 8: $D_4$

Order 7: $C_7$

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

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 504: $C_7:C_3\times S_4$ ($C_1$)

Order 252: $A_4\times C_7:C_3$ ($C_2$)

Order 168: $C_7\times S_4$ ($C_3$)

Order 84: $C_{14}:C_6$ ($S_3$), $C_7\times A_4$ ($C_6$)

Order 28: $C_2\times C_{14}$ ($C_3\times S_3$)

Order 24: $S_4$ ($C_7:C_3$)

Order 21: $C_7:C_3$ ($S_4$)

Order 12: $A_4$ ($C_7:C_6$)

Order 7: $C_7$ ($C_3\times S_4$)

Order 4: $C_2^2$ ($C_{21}:C_6$)

Order 1: $C_1$ ($C_7:C_3\times S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 504: $C_7:C_3\times S_4$ ($C_1$)

Order 252: $A_4\times C_7:C_3$ ($C_2$)

Order 168: $C_7\times S_4$ ($C_3$)

Order 84: $C_{14}:C_6$ ($S_3$), $C_7\times A_4$ ($C_6$)

Order 28: $C_2\times C_{14}$ ($C_3\times S_3$)

Order 24: $S_4$ ($C_7:C_3$)

Order 21: $C_7:C_3$ ($S_4$)

Order 12: $A_4$ ($C_7:C_6$)

Order 7: $C_7$ ($C_3\times S_4$)

Order 4: $C_2^2$ ($C_{21}:C_6$)

Order 1: $C_1$ ($C_7:C_3\times S_4$)

Series

Derived series	$C_7:C_3\times S_4$	$\rhd$	$C_7\times A_4$	$\rhd$	$C_2^2$	$\rhd$	$C_1$
Chief series	$C_7:C_3\times S_4$	$\rhd$	$A_4\times C_7:C_3$	$\rhd$	$C_7\times A_4$	$\rhd$	$C_2\times C_{14}$	$\rhd$	$C_2^2$	$\rhd$	$C_1$
Lower central series	$C_7:C_3\times S_4$	$\rhd$	$C_7\times A_4$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 15 larger groups in the database.

This group is a maximal quotient of 18 larger groups in the database.

Character theory

Complex character table

See the $25 \times 25$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	3A	3B	3C	4A	6A	6B	7A	12A	14A	14B	21A	28A
	Size	1	3	6	14	8	112	6	42	84	6	84	18	36	48	36
	2 P	1A	1A	1A	3A	3B	3C	2A	3A	3A	7A	6A	7A	7A	21A	14A
	3 P	1A	2A	2B	1A	1A	1A	4A	2A	2B	7A	4A	14A	14B	7A	28A
	7 P	1A	2A	2B	3A	3B	3C	4A	6A	6B	1A	12A	2A	2B	3B	4A
504.159.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
504.159.1b		$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$
504.159.1c		$2$	$2$	$2$	$-1$	$2$	$-1$	$2$	$-1$	$-1$	$2$	$-1$	$2$	$2$	$2$	$2$
504.159.1d		$2$	$2$	$-2$	$-1$	$2$	$-1$	$-2$	$-1$	$1$	$2$	$1$	$2$	$-2$	$2$	$-2$
504.159.2a		$2$	$2$	$0$	$2$	$-1$	$-1$	$0$	$2$	$0$	$2$	$0$	$2$	$0$	$-1$	$0$
504.159.2b		$4$	$4$	$0$	$-2$	$-2$	$1$	$0$	$-2$	$0$	$4$	$0$	$4$	$0$	$-2$	$0$
504.159.3a		$3$	$-1$	$1$	$3$	$0$	$0$	$-1$	$-1$	$1$	$3$	$-1$	$-1$	$1$	$0$	$-1$
504.159.3b		$3$	$-1$	$-1$	$3$	$0$	$0$	$1$	$-1$	$-1$	$3$	$1$	$-1$	$-1$	$0$	$1$
504.159.3c		$6$	$6$	$6$	$0$	$6$	$0$	$6$	$0$	$0$	$-1$	$0$	$-1$	$-1$	$-1$	$-1$
504.159.3d		$6$	$-2$	$2$	$-3$	$0$	$0$	$-2$	$1$	$-1$	$6$	$1$	$-2$	$2$	$0$	$-2$
504.159.3e		$6$	$6$	$-6$	$0$	$6$	$0$	$-6$	$0$	$0$	$-1$	$0$	$-1$	$1$	$-1$	$1$
504.159.3f		$6$	$-2$	$-2$	$-3$	$0$	$0$	$2$	$1$	$1$	$6$	$-1$	$-2$	$-2$	$0$	$2$
504.159.6a		$12$	$12$	$0$	$0$	$-6$	$0$	$0$	$0$	$0$	$-2$	$0$	$-2$	$0$	$1$	$0$
504.159.9a		$18$	$-6$	$6$	$0$	$0$	$0$	$-6$	$0$	$0$	$-3$	$0$	$1$	$-1$	$0$	$1$
504.159.9b		$18$	$-6$	$-6$	$0$	$0$	$0$	$6$	$0$	$0$	$-3$	$0$	$1$	$1$	$0$	$-1$