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C₄×C₇⋊C₈ #6376

order	224
isoclinism	isoclinic class of D₇
automorphism	Aut(G) = AGL(1,𝔽₇)×#2021 (order 5376) · Inn(G) = D₇ · Out(G) = C₃×#2021
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄

Aliases

C₄×C₇⋊C₈

Splittings

C₇ ⋊ C₄×C₈

Subgroups

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

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×C₇⋊C₈, #89249, #89250, …
maximal normal subgroup of	C₈×C₇⋊C₈, #89249, #89250, …
maximal quotient of	C₈×C₇⋊C₈, #89250, C₄×C₇⋊C₁₆, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 8);