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D₈×Dic₇ #89921

order	448
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

D₈×Dic₇

Splittings

C₇ ⋊ C₄×D₈

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₇
commutator	C₄×C₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₇
Fitting, prime radical	C₂×C₇×D₈	C₂
socle	C₂²×C₇	C₂×D₄
cosocle	C₂×C₄×C₇	C₂³
hypercenter	C₂×D₈	D₇
hypocenter	C₇	C₄×D₈
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 224

A minimal faithful permutation representation — 3 generators on 19 points:

(2,3)(4,5)(6,7)(8,9,10,11);(1,2,4,6,7,5,3)(13,16)(14,19)(17,18);(12,13)(14,16)(15,18)(17,19)

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11726928, #11726929, #11726930, …
maximal normal subgroup of	#11726928, #11726929, #11726930, …
maximal quotient of	Dic₇×#102, #11727909, #11727922, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 683);