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C₄×D₄₀ #64981

order	320
isoclinism	isoclinic class of #677
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

C₄×D₄₀

Splittings

Fitting	C₄×C₅×C₈ ⋊ C₂
Sylow	C₅ ⋊ C₄×D₈

Subgroups

	subgroup	G/H
center	C₂×C₄	D₂₀
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₄×C₅×C₈	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂×D₄₀	C₂
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₄×C₈	D₅
hypocenter	C₅	C₄×D₈
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10597956, #10597962, #10597995, …
maximal normal subgroup of	#10597956, #10597962, #10597995, …
maximal quotient of	C₈×D₄₀, #10598845, #10598871, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 319);