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C₅×#713 #87894

order	400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂²

Aliases

Splittings

Fitting	C₅²×Q₈ ⋊ C₂
Sylow	C₅² ⋊ #41

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂²×D₅
commutator, cosocle	C₂×C₅	C₂³×C₅
Frattini	C₂	C₂²×C₅×D₅
Fitting	C₅²×Q₈	C₂
socle	C₂×C₅²	C₂³
hypercenter	C₅×Q₈	D₅
hypocenter	C₅	C₅×#41
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅ ◁ C₅×Q₈

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 18 points:

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

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅×#4010, #11716853, #11717057, …
maximal normal subgroup of	C₅×#4010, #11716853, #11717057, …
maximal quotient of	C₅×#4036, C₅×#4038, C₅×#4039, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 188);