Abstract group 672.1049: $F_8:C_{12}$

• Label: 672.1049
• Order: \( 2^{5} \cdot 3 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $12$
• Trans deg.: $28$
• Rank: $2$

Group information

Description:	$F_8:C_{12}$
Order:	\(672\)\(\medspace = 2^{5} \cdot 3 \cdot 7 \)
Exponent:	\(84\)\(\medspace = 2^{2} \cdot 3 \cdot 7 \)
Automorphism group:	$F_8:C_6$, of order \(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 5, $C_3$, $C_7$
Derived length:	$3$

This group is nonabelian, monomial (hence solvable), and an A-group.

Group statistics

Order	1	2	3	4	6	7	12	14	28
Elements	1	15	56	16	168	48	224	48	96	672
Conjugacy classes	1	3	2	4	6	2	8	2	4	32
Divisions	1	3	1	2	3	1	2	1	1	15
Autjugacy classes	1	3	2	2	6	2	4	2	2	24

Dimension	1	2	3	4	6	7	12	14	28
Irr. complex chars.	12	0	8	0	0	12	0	0	0	32
Irr. rational chars.	2	3	0	1	2	2	1	3	1	15

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$28$
Rank:	$2$
Inequivalent generating pairs:	$672$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	7	14	14
Arbitrary	7	9	9

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{3}=b^{28}=c^{2}=d^{2}=e^{2}=[c,d]=[c,e]=[d,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $12$ $\langle(9,10,11,12), (2,4,6)(3,7,8), (9,11)(10,12), (2,5,7,8,6,3,4), (1,2)(3,5)(4,6)(7,8), (1,3)(2,5)(4,8)(6,7), (1,4)(2,6)(3,8)(5,7)\rangle$
Transitive group:	28T79				more information
Direct product:	$C_4$ $\, \times\, $ $(F_8:C_3)$
Semidirect product:	$F_8$ $\,\rtimes\,$ $C_{12}$	$(C_4\times F_8)$ $\,\rtimes\,$ $C_3$	$C_2^3$ $\,\rtimes\,$ $(C_7:C_{12})$	$(C_2^3\times C_4)$ $\,\rtimes\,$ $(C_7:C_3)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(F_8:C_6)$ . $C_2$	$(C_2\times F_8)$ . $C_6$	$C_2$ . $(F_8:C_6)$	$C_2^4$ . $(C_7:C_6)$	more information
Aut. group:	$\Aut(C_5\times F_8)$	$\Aut(C_{10}\times F_8)$	$\Aut(F_8:C_{15})$	$\Aut(F_8:C_{30})$

Elements of the group are displayed as permutations of degree 12.

Homology

Abelianization:	$C_{12} \simeq C_{4} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 452 subgroups in 46 conjugacy classes, 12 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $F_8:C_3$
Commutator:	$G' \simeq$ $F_8$	$G/G' \simeq$ $C_{12}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $F_8:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^3\times C_4$	$G/\operatorname{Fit} \simeq$ $C_7:C_3$
Radical:	$R \simeq$ $F_8:C_{12}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_7:C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3\times C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
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 672: $F_8:C_{12}$

Order 336: $F_8:C_6$

Order 224: $C_4\times F_8$

Order 168: $F_8:C_3$

Order 112: $C_2\times F_8$

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

Order 84: $C_7:C_{12}$

Order 56: $F_8$

Order 48: $C_4\times A_4$ x 2, $C_2^2\times A_4$

Order 42: $C_7:C_6$

Order 32: $C_2^3\times C_4$

Order 28: $C_{28}$

Order 24: $C_2\times A_4$ x 3, $C_2\times C_{12}$

Order 21: $C_7:C_3$

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

Order 14: $C_{14}$

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

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

Order 7: $C_7$

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 672: $F_8:C_{12}$

Order 336: $F_8:C_6$

Order 224: $C_4\times F_8$

Order 168: $F_8:C_3$

Order 112: $C_2\times F_8$

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

Order 84: $C_7:C_{12}$

Order 56: $F_8$

Order 48: $C_4\times A_4$ x 2, $C_2^2\times A_4$

Order 42: $C_7:C_6$

Order 32: $C_2^3\times C_4$

Order 28: $C_{28}$

Order 24: $C_2\times A_4$ x 3, $C_2\times C_{12}$

Order 21: $C_7:C_3$

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

Order 14: $C_{14}$

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

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

Order 7: $C_7$

Order 6: $C_6$ x 3

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 672: $F_8:C_{12}$ ($C_1$)

Order 336: $F_8:C_6$ ($C_2$)

Order 224: $C_4\times F_8$ ($C_3$)

Order 168: $F_8:C_3$ ($C_4$)

Order 112: $C_2\times F_8$ ($C_6$)

Order 56: $F_8$ ($C_{12}$)

Order 32: $C_2^3\times C_4$ ($C_7:C_3$)

Order 16: $C_2^4$ ($C_7:C_6$)

Order 8: $C_2^3$ ($C_7:C_{12}$)

Order 4: $C_4$ ($F_8:C_3$)

Order 2: $C_2$ ($F_8:C_6$)

Order 1: $C_1$ ($F_8:C_{12}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 672: $F_8:C_{12}$ ($C_1$)

Order 336: $F_8:C_6$ ($C_2$)

Order 224: $C_4\times F_8$ ($C_3$)

Order 168: $F_8:C_3$ ($C_4$)

Order 112: $C_2\times F_8$ ($C_6$)

Order 56: $F_8$ ($C_{12}$)

Order 32: $C_2^3\times C_4$ ($C_7:C_3$)

Order 16: $C_2^4$ ($C_7:C_6$)

Order 8: $C_2^3$ ($C_7:C_{12}$)

Order 4: $C_4$ ($F_8:C_3$)

Order 2: $C_2$ ($F_8:C_6$)

Order 1: $C_1$ ($F_8:C_{12}$)

Series

Derived series

$F_8:C_{12}$

$\rhd$

$F_8$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Chief series

$F_8:C_{12}$

$\rhd$

$F_8:C_6$

$\rhd$

$F_8:C_3$

$\rhd$

$F_8$

$\rhd$

$C_2^3$

$\rhd$

$C_1$

Lower central series

$F_8:C_{12}$

$\rhd$

$F_8$

Upper central series

$C_1$

$\lhd$

$C_4$

Supergroups

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

This group is a maximal quotient of 11 larger groups in the database.

Character theory

Complex character table

See the $32 \times 32$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	3A	4A	4B	6A	6B	6C	7A	12A	12B	14A	28A
	Size	1	1	7	7	56	2	14	56	56	56	48	112	112	48	96
	2 P	1A	1A	1A	1A	3A	2A	2A	3A	3A	3A	7A	6B	6B	7A	14A
	3 P	1A	2A	2B	2C	1A	4A	4B	2B	2A	2C	7A	4B	4A	14A	28A
	7 P	1A	2A	2B	2C	3A	4A	4B	6A	6B	6C	1A	12A	12B	2A	4A
672.1049.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
672.1049.1b		$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$
672.1049.1c		$2$	$2$	$2$	$2$	$-1$	$2$	$2$	$-1$	$-1$	$-1$	$2$	$-1$	$-1$	$2$	$2$
672.1049.1d		$2$	$-2$	$-2$	$2$	$2$	$0$	$0$	$-2$	$-2$	$2$	$2$	$0$	$0$	$-2$	$0$
672.1049.1e		$2$	$2$	$2$	$2$	$-1$	$-2$	$-2$	$-1$	$-1$	$-1$	$2$	$1$	$1$	$2$	$-2$
672.1049.1f		$4$	$-4$	$-4$	$4$	$-2$	$0$	$0$	$2$	$2$	$-2$	$4$	$0$	$0$	$-4$	$0$
672.1049.3a		$6$	$6$	$6$	$6$	$0$	$6$	$6$	$0$	$0$	$0$	$-1$	$0$	$0$	$-1$	$-1$
672.1049.3b		$6$	$6$	$6$	$6$	$0$	$-6$	$-6$	$0$	$0$	$0$	$-1$	$0$	$0$	$-1$	$1$
672.1049.3c		$12$	$-12$	$-12$	$12$	$0$	$0$	$0$	$0$	$0$	$0$	$-2$	$0$	$0$	$2$	$0$
672.1049.7a		$7$	$7$	$-1$	$-1$	$1$	$7$	$-1$	$-1$	$1$	$-1$	$0$	$-1$	$1$	$0$	$0$
672.1049.7b		$7$	$7$	$-1$	$-1$	$1$	$-7$	$1$	$-1$	$1$	$-1$	$0$	$1$	$-1$	$0$	$0$
672.1049.7c		$14$	$14$	$-2$	$-2$	$-1$	$14$	$-2$	$1$	$-1$	$1$	$0$	$1$	$-1$	$0$	$0$
672.1049.7d		$14$	$-14$	$2$	$-2$	$2$	$0$	$0$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$
672.1049.7e		$14$	$14$	$-2$	$-2$	$-1$	$-14$	$2$	$1$	$-1$	$1$	$0$	$-1$	$1$	$0$	$0$
672.1049.7f		$28$	$-28$	$4$	$-4$	$-2$	$0$	$0$	$-2$	$2$	$2$	$0$	$0$	$0$	$0$	$0$