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C₂×#4021 #65810

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

Aliases

Splittings

Fitting	C₂×C₄²×C₅ ⋊ C₂
Sylow	C₅ ⋊ C₂×#124

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁴
Frattini	C₂²	C₂³×D₅
Fitting	C₂×C₄²×C₅	C₂
socle	C₂³×C₅	C₂³
prime radical	C₂²×D₂₀	C₂
hypercenter	C₂×C₄²	D₅
hypocenter	C₅	C₂×#124
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 4 generators on 19 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10599423, #10599468, #10599470, …
maximal normal subgroup of	#10599423, #10599468, #10599470, …
maximal quotient of	C₄×#4021, #10605039, #10605044, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1148);