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D₈×AGL(1,𝔽₅) #65730

order	320
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

D₈×AGL(1,𝔽₅)

Splittings

Fitting	C₅×D₈ ⋊ C₄
Sylow	C₅ ⋊ C₄×D₈

Subgroups

	subgroup	G/H
center	C₂	D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₄
Frattini	C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×D₈	C₄
socle	C₂×C₅	C₄×D₄
prime radical	D₅×D₈	C₂
cosocle	C₄×D₅	C₂³
hypercenter	D₈	AGL(1,𝔽₅)
hypocenter	C₅	C₄×D₈
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 40

A minimal faithful permutation representation — 3 generators on 13 points:

(2,3,4,5);(1,2,3,5,4)(7,10)(8,13)(11,12);(6,7)(8,10)(9,12)(11,13)

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₅)×D₁₆, #10604121, AGL(1,𝔽₅)×SD₃₂, …
maximal normal subgroup of	AGL(1,𝔽₅)×D₁₆, #10604121, AGL(1,𝔽₅)×SD₃₂, …
maximal quotient of	#10603935, #10603939, #10603940, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1068);