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C₅²×Dic₅ #92551

order	500
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₄×C₅² · Schur multiplier (H₂) = C₅

Aliases

C₅²×Dic₅

Splittings

derived	C₅ ⋊ C₄×C₅²
Sylow	C₅³ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅²	D₅
commutator, hypocenter	C₅	C₄×C₅²
Frattini	C₂	C₅²×D₅
Fitting, socle, prime radical	C₂×C₅³	C₂
cosocle	C₂×C₅	C₂×C₅²
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₅²

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 100

A minimal faithful permutation representation — 2 generators on 19 points:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅²×#170, C₅×#6032, #11758739, …
maximal normal subgroup of	C₅²×#170, C₅×#6032, #11758739, …
maximal quotient of	C₅²×#168, C₂×C₅²×Dic₅, C₃×C₅²×Dic₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(500, 37);