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C₄²×Dic₅ #65219

order	320
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₄³

Aliases

C₄²×Dic₅

Splittings

C₅ ⋊ C₄³

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄²	D₅
commutator, hypocenter	C₅	C₄³
Frattini	C₂³	C₂²×D₅
Fitting	C₂×C₄²×C₅	C₂
socle, prime radical, cosocle	C₂³×C₅	C₂³
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₄²

258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 17 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598227, #10598696, #10599389, …
maximal normal subgroup of	#10598227, #10598696, #10599389, …
maximal quotient of	C₄²×#168, C₄×#3935, Dic₅×#95, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 557);