Abstract group 448.1363: $D_4\times F_8$

• Label: 448.1363
• Order: \( 2^{6} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 7 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \)
• Perm deg.: $12$
• Trans deg.: $28$
• Rank: $2$

Group information

Description:	$D_4\times F_8$
Order:	\(448\)\(\medspace = 2^{6} \cdot 7 \)
Exponent:	\(28\)\(\medspace = 2^{2} \cdot 7 \)
Automorphism group:	$D_4\times F_8:C_3$, of order \(1344\)\(\medspace = 2^{6} \cdot 3 \cdot 7 \)
Composition factors:	$C_2$ x 6, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	4	7	14	28
Elements	1	47	16	48	240	96	448
Conjugacy classes	1	7	2	6	18	6	40
Divisions	1	7	2	1	3	1	15
Autjugacy classes	1	5	2	2	4	2	16

Dimension	1	2	6	7	12	14
Irr. complex chars.	28	7	0	4	0	1	40
Irr. rational chars.	4	1	4	4	1	1	15

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$28$
Rank:	$2$
Inequivalent generating pairs:	$48$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c, d, e \mid a^{2}=b^{28}=c^{2}=d^{2}=e^{2}=[a,c]=[a,d]=[a,e]=[c,d]=[c,e]=[d,e]=1, b^{a}=b^{15}, c^{b}=cde, d^{b}=cd, e^{b}=de \rangle$
Permutation group:	Degree $12$ $\langle(9,10)(11,12), (10,12), (2,3,4,6,8,5,7), (9,11)(10,12), (1,2)(3,5)(4,6)(7,8), (1,3)(2,5)(4,7)(6,8), (1,4)(2,6)(3,7)(5,8)\rangle$
Transitive group:	28T59	32T9549			more information
Direct product:	$D_4$ $\, \times\, $ $F_8$
Semidirect product:	$C_2^5$ $\,\rtimes\,$ $C_{14}$ (2)	$(C_4\times F_8)$ $\,\rtimes\,$ $C_2$	$C_4$ $\,\rtimes\,$ $(C_2\times F_8)$	$(D_4\times C_2^3)$ $\,\rtimes\,$ $C_7$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times F_8)$ . $C_2^2$	$C_2$ . $(C_2^2\times F_8)$	$C_2^4$ . $(C_2\times C_{14})$		more information

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

Homology

Abelianization:	$C_{2} \times C_{14} \simeq C_{2}^{2} \times C_{7}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 1027 subgroups in 127 conjugacy classes, 18 normal (12 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^2\times F_8$
Commutator:	$G' \simeq$ $C_2^4$	$G/G' \simeq$ $C_2\times C_{14}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^2\times F_8$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_2^3$	$G/\operatorname{Fit} \simeq$ $C_7$
Radical:	$R \simeq$ $D_4\times F_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_2\times C_{14}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4\times C_2^3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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

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Classes of subgroups up to conjugation

Order 448: $D_4\times F_8$

Order 224: $C_2^2\times F_8$ x 2, $C_4\times F_8$

Order 112: $C_2\times F_8$ x 3

Order 64: $D_4\times C_2^3$

Order 56: $C_7\times D_4$, $F_8$

Order 32: $C_2^2\times D_4$ x 4, $C_2^5$ x 2, $C_2^3\times C_4$

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

Order 16: $C_2\times D_4$ x 16, $C_2^4$ x 9, $C_2^2\times C_4$ x 2

Order 14: $C_{14}$ x 3

Order 8: $C_2^3$ x 27, $D_4$ x 10, $C_2\times C_4$ x 4

Order 7: $C_7$

Order 4: $C_2^2$ x 25, $C_4$ x 2

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 448: $D_4\times F_8$

Order 224: $C_2^2\times F_8$, $C_4\times F_8$

Order 112: $C_2\times F_8$ x 2

Order 64: $D_4\times C_2^3$

Order 56: $C_7\times D_4$, $F_8$

Order 32: $C_2^2\times D_4$ x 3, $C_2^5$, $C_2^3\times C_4$

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

Order 16: $C_2^4$ x 5, $C_2\times D_4$ x 4, $C_2^2\times C_4$ x 2

Order 14: $C_{14}$ x 2

Order 8: $C_2^3$ x 9, $D_4$ x 4, $C_2\times C_4$ x 2

Order 7: $C_7$

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 448: $D_4\times F_8$ ($C_1$)

Order 224: $C_2^2\times F_8$ ($C_2$) x 2, $C_4\times F_8$ ($C_2$)

Order 112: $C_2\times F_8$ ($C_2^2$)

Order 64: $D_4\times C_2^3$ ($C_7$)

Order 56: $F_8$ ($D_4$)

Order 32: $C_2^5$ ($C_{14}$) x 2, $C_2^3\times C_4$ ($C_{14}$)

Order 16: $C_2^4$ ($C_2\times C_{14}$)

Order 8: $C_2^3$ ($C_7\times D_4$), $D_4$ ($F_8$)

Order 4: $C_2^2$ ($C_2\times F_8$) x 2, $C_4$ ($C_2\times F_8$)

Order 2: $C_2$ ($C_2^2\times F_8$)

Order 1: $C_1$ ($D_4\times F_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 448: $D_4\times F_8$ ($C_1$)

Order 224: $C_2^2\times F_8$ ($C_2$), $C_4\times F_8$ ($C_2$)

Order 112: $C_2\times F_8$ ($C_2^2$)

Order 64: $D_4\times C_2^3$ ($C_7$)

Order 56: $F_8$ ($D_4$)

Order 32: $C_2^5$ ($C_{14}$), $C_2^3\times C_4$ ($C_{14}$)

Order 16: $C_2^4$ ($C_2\times C_{14}$)

Order 8: $C_2^3$ ($C_7\times D_4$), $D_4$ ($F_8$)

Order 4: $C_2^2$ ($C_2\times F_8$), $C_4$ ($C_2\times F_8$)

Order 2: $C_2$ ($C_2^2\times F_8$)

Order 1: $C_1$ ($D_4\times F_8$)

Series

Derived series

$D_4\times F_8$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$D_4\times F_8$

$\rhd$

$C_2^2\times F_8$

$\rhd$

$C_2\times F_8$

$\rhd$

$C_2^4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$D_4\times F_8$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$D_4$

Supergroups

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

This group is a maximal quotient of 9 larger groups in the database.

Character theory

Complex character table

See the $40 \times 40$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	2D	2E	2F	2G	4A	4B	7A	14A	14B	14C	28A
	Size	1	1	2	2	7	7	14	14	2	14	48	48	96	96	96
	2 P	1A	1A	1A	1A	1A	1A	1A	1A	2A	2A	7A	7A	7A	7A	14A
	7 P	1A	2A	2B	2C	2D	2E	2F	2G	4A	4B	7A	14A	14B	14C	28A
448.1363.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
448.1363.1b		$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$
448.1363.1c		$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$
448.1363.1d		$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$
448.1363.1e		$6$	$6$	$6$	$6$	$6$	$6$	$6$	$6$	$6$	$6$	$-1$	$-1$	$-1$	$-1$	$-1$
448.1363.1f		$6$	$6$	$-6$	$-6$	$6$	$6$	$-6$	$-6$	$6$	$6$	$-1$	$-1$	$1$	$1$	$-1$
448.1363.1g		$6$	$6$	$-6$	$6$	$6$	$6$	$-6$	$6$	$-6$	$-6$	$-1$	$-1$	$1$	$-1$	$1$
448.1363.1h		$6$	$6$	$6$	$-6$	$6$	$6$	$6$	$-6$	$-6$	$-6$	$-1$	$-1$	$-1$	$1$	$1$
448.1363.2a		$2$	$-2$	$0$	$0$	$2$	$-2$	$0$	$0$	$0$	$0$	$2$	$-2$	$0$	$0$	$0$
448.1363.2b		$12$	$-12$	$0$	$0$	$12$	$-12$	$0$	$0$	$0$	$0$	$-2$	$2$	$0$	$0$	$0$
448.1363.7a		$7$	$7$	$7$	$7$	$-1$	$-1$	$-1$	$-1$	$7$	$-1$	$0$	$0$	$0$	$0$	$0$
448.1363.7b		$7$	$7$	$-7$	$-7$	$-1$	$-1$	$1$	$1$	$7$	$-1$	$0$	$0$	$0$	$0$	$0$
448.1363.7c		$7$	$7$	$-7$	$7$	$-1$	$-1$	$1$	$-1$	$-7$	$1$	$0$	$0$	$0$	$0$	$0$
448.1363.7d		$7$	$7$	$7$	$-7$	$-1$	$-1$	$-1$	$1$	$-7$	$1$	$0$	$0$	$0$	$0$	$0$
448.1363.14a		$14$	$-14$	$0$	$0$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$