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C₂×C₅×#168 #87787

order	400
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₈ · Schur multiplier (H₂) = C₂

Aliases

C₂×C₅×#168

Splittings

derived	C₅ ⋊ C₂×C₅×C₈
Sylow	C₅² ⋊ C₂×C₈

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄×C₅	D₅
commutator, hypocenter	C₅	C₂×C₅×C₈
Frattini	C₄	C₂×C₅×D₅
Fitting	C₂×C₄×C₅²	C₂
socle, prime radical	C₂²×C₅²	C₄
cosocle	C₄×C₅	C₂²×C₅
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 2 generators on 20 points:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	Dic₅×#168, #11716594, #11716595, …
maximal normal subgroup of	Dic₅×#168, #11716594, #11716595, …
maximal quotient of	C₄×C₅×#168, C₅×C₄⋊#168, C₂×C₅×#672, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 81);