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C₂×#4086 #66137

order	320
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

Splittings

Fitting	C₂²×C₅×D₄ ⋊ C₂
Sylow	C₅ ⋊ C₂×#127

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁴
Frattini	C₂²	C₂³×D₅
Fitting	C₂²×C₅×D₄	C₂
socle	C₂³×C₅	C₂³
hypercenter	C₂²×D₄	D₅
hypocenter	C₅	C₂×#127
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#64694, #10600857, #10600863, …
maximal normal subgroup of	C₂×#64694, #10600857, #10600863, …
maximal quotient of	C₂×#65246, C₂×#65238, #10613846, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1475);