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C₈×ΓL(1,𝔽₈)⋊D₇ #72245166

order	2352
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂

Aliases

C₈×ΓL(1,𝔽₈)⋊D₇

Splittings

Fitting	C₇²×C₈ ⋊ C₂×C₃
derived, Sylow	C₇² ⋊ C₂×C₃×C₈
Hall	C₈×C₇⋊D₇ ⋊ C₃
Hall	C₇×ΓL(1,𝔽₈) ⋊ C₂×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₈	ΓL(1,𝔽₈)⋊D₇
commutator, hypocenter	C₇²	C₂×C₃×C₈
Frattini	C₄	C₂×ΓL(1,𝔽₈)⋊D₇
Fitting	C₇²×C₈	C₂×C₃
socle	C₂×C₇²	C₂×C₃×C₄
prime radical	C₂×ΓL(1,𝔽₈)⋊D₇	C₄
cosocle	C₄×C₇²	C₂²×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₈

110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	M₁₆×ΓL(1,𝔽₈)⋊D₇, SD₁₆×ΓL(1,𝔽₈)⋊D₇, D₈×ΓL(1,𝔽₈)⋊D₇, …
maximal normal subgroup of	M₁₆×ΓL(1,𝔽₈)⋊D₇, SD₁₆×ΓL(1,𝔽₈)⋊D₇, D₈×ΓL(1,𝔽₈)⋊D₇, …

GAP

Past the SmallGroups library coverage — no SmallGroups id.