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C₂×#6045 #87897

order	400
isoclinism	isoclinic class of #6045
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₅

Aliases

Splittings

C₅² ⋊ C₂×Q₈

Subgroups

	subgroup	G/H
center	C₂²	C₂×C₅⋊D₅
commutator, cosocle	C₂×C₅²	C₂³
Frattini	C₂	C₂²×C₅⋊D₅
Fitting	C₂×C₄×C₅²	C₂
socle, prime radical	C₂²×C₅²	C₂²
hypercenter	C₂×C₄	C₅⋊D₅
hypocenter	C₅²	C₂×Q₈
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 400

A minimal faithful permutation representation — 3 generators on 20 points:

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

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11716602, #11716608, #11716677, …
maximal normal subgroup of	#11716602, #11716608, #11716677, …
maximal quotient of	C₄×#6045, #11717189, #11717190, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 191);