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C₂×#3967 #65513

order	320
isoclinism	isoclinic class of #686
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

C₅ ⋊ C₂×#103

Subgroups

	subgroup	G/H
center	C₂³	C₂²⋊D₅
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₂²×C₅×Q₈	C₂
socle, prime radical	C₂³×C₅	D₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₂²×Q₈	D₅
hypocenter	C₅	C₂×#103
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 19 points:

(7,8,10,11,9)(12,13,15,17)(14,19,18,16);(3,4,5,6)(8,9)(10,11)(13,16)(14,18)(17,19);(1,2)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10600459, #10600460, #10600461, …
maximal normal subgroup of	#10600459, #10600460, #10600461, …
maximal quotient of	#10602347, C₂×#64749, C₂×Q₈⋊#168, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 851);