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C₂²×Dic₂₀ #66076

order	320
isoclinism	isoclinic class of #677
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂²×Dic₂₀

Splittings

C₅ ⋊ C₂²×Q₁₆

Subgroups

	subgroup	G/H
center	C₂³	D₂₀
commutator, cosocle	C₄×C₅	C₂⁴
Frattini	C₄	C₂³×D₅
Fitting	C₂²×C₅×C₈	C₂
socle, prime radical	C₂³×C₅	D₄
hypercenter	C₂²×C₈	D₅
hypocenter	C₅	C₂²×Q₁₆
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 25 points:

(5,6,8,9,7)(10,12,19,23,14,21,13,20)(11,15,22,25,17,24,16,18);(1,2)(3,4)(6,7)(8,9)(10,11,14,17)(12,18,21,25)(13,22,19,16)(15,23,24,20);(3,4);(1,2)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10599470, #10599478, #10599481, …
maximal normal subgroup of	#10599470, #10599478, #10599481, …
maximal quotient of	C₂×#64971, C₂×C₄×Dic₂₀, C₂×#64970, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1414);