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C₂³×C₄×D₅ #66271

order	320
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

C₂³×C₄×D₅

Splittings

Fitting	C₂³×C₄×C₅ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂⁴×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂³×C₄	D₅
commutator, hypocenter	C₅	C₂⁴×C₄
Frattini	C₂	C₂⁴×D₅
Fitting	C₂³×C₄×C₅	C₂
socle	C₂⁴×C₅	C₂²
prime radical	C₂⁴×D₅	C₂
cosocle	C₂×C₅	C₂⁵
LCS	G ▷ C₅
UCS	C₁ ◁ C₂³×C₄

1362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 15 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10600218, #10600748, #10604898, …
maximal normal subgroup of	#10600218, #10600748, #10604898, …
maximal quotient of	C₂²×C₄×Dic₁₀, C₂×C₄×#709, C₂²×C₄²×D₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1609);