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C₂²×#686 #66126

order	320
isoclinism	isoclinic class of #686
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

C₂²×#686

Splittings

Fitting	C₂²×C₅×D₄ ⋊ C₂
Sylow	C₅ ⋊ C₂²×D₈

Subgroups

	subgroup	G/H
center	C₂³	C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂⁴
Frattini	C₄	C₂³×D₅
Fitting	C₂²×C₅×D₄	C₂
socle	C₂³×C₅	D₄
hypercenter	C₂²×D₄	D₅
hypocenter	C₅	C₂²×D₈
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×D₄

338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10600440, #10600441, #10600443, …
maximal normal subgroup of	#10600440, #10600441, #10600443, …
maximal quotient of	C₂²×#3939, C₂²×#3941, C₂×#65256, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1464);