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C₂²×#711 #66275

order	320
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

C₂²×#711

Splittings

Fitting	C₂²×C₅×D₄ ⋊ C₂
Sylow	C₅ ⋊ C₂²×#41

Subgroups

	subgroup	G/H
center	C₂³	C₂²×D₅
commutator, cosocle	C₂×C₅	C₂⁵
Frattini	C₂	C₂⁴×D₅
Fitting	C₂²×C₅×D₄	C₂
socle	C₂³×C₅	C₂³
hypercenter	C₂²×D₄	D₅
hypocenter	C₅	C₂²×#41
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 17 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10600820, #10600873, #10600992, …
maximal normal subgroup of	#10600820, #10600873, #10600992, …
maximal quotient of	C₂²×#4023, C₂²×#4027, C₂²×#4024, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1613);