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C₂×C₄²×D₅ #65805

order	320
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

C₂×C₄²×D₅

Splittings

Fitting	C₂×C₄²×C₅ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂²×C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄²	D₅
commutator, hypocenter	C₅	C₂²×C₄²
Frattini	C₂²	C₂³×D₅
Fitting	C₂×C₄²×C₅	C₂
socle	C₂³×C₅	C₂³
prime radical	C₂³×D₅	C₂²
cosocle	C₂²×C₅	C₂⁴
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₄²

498 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)(12,13)(14,15);(3,4,5,6)(11,12,14,15,13);(7,8,9,10);(1,2)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10599392, #10599393, D₅×#336, …
maximal normal subgroup of	#10599392, #10599393, D₅×#336, …
maximal quotient of	C₄×#4018, C₄²×D₂₀, C₄²×Dic₁₀, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1143);