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C₈×C₇⋊C₈ #89248

order	448
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₈

Aliases

direct product	C₈×C₇⋊C₈
unambiguous semidirect product	C₇⋊C₈²

Splittings

C₇ ⋊ C₈²

Subgroups

	subgroup	G/H
center, hypercenter	C₄×C₈	D₇
commutator, hypocenter	C₇	C₈²
Frattini	C₄²	C₂×D₇
Fitting	C₄×C₇×C₈	C₂
socle, prime radical	C₂²×C₇	C₄²
cosocle	C₄²×C₇	C₂²
LCS	G ▷ C₇
UCS	C₁ ◁ C₄×C₈

74 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 2 generators on 23 points:

(2,3)(4,5)(6,7)(16,17,18,20,19,21,22,23);(1,2,4,6,7,5,3)(8,9,10,12,11,13,14,15)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₁₆×C₇⋊C₈, #11725304, #11725305, …
maximal normal subgroup of	C₁₆×C₇⋊C₈, #11725304, #11725305, …
maximal quotient of	C₈×C₇⋊C₁₆, #11725243, C₁₆×C₇⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 10);