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C₂³×Dic₅ #4151

order	160
isoclinism	isoclinic class of D₅
automorphism	Aut(G) order 430080
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂³×Dic₅

Splittings

C₅ ⋊ C₂³×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂⁴	D₅
commutator, hypocenter	C₅	C₂³×C₄
Frattini	C₂	C₂³×D₅
Fitting, socle, prime radical	C₂⁴×C₅	C₂
cosocle	C₂×C₅	C₂⁴
LCS	G ▷ C₅
UCS	C₁ ◁ C₂⁴

236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 4 generators on 15 points:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²⋊C₄×Dic₅, #65231, #65232, …
maximal normal subgroup of	C₂²⋊C₄×Dic₅, #65231, #65232, …
maximal quotient of	C₂³×#168, C₂²×#681, C₂²×C₄×Dic₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(160, 226);