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C₂²×#689 #66143

order	320
isoclinism	isoclinic class of #686
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂²×#689

Splittings

C₅ ⋊ C₂²×Q₁₆

Subgroups

	subgroup	G/H
center	C₂³	C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂⁴
Frattini	C₄	C₂³×D₅
Fitting	C₂²×C₅×Q₈	C₂
socle, prime radical	C₂³×C₅	D₄
hypercenter	C₂²×Q₈	D₅
hypocenter	C₅	C₂²×Q₁₆
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

258 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 4 generators on 25 points:

(5,6,8,9,7)(10,12,14,20)(11,15,17,24)(13,22,23,19)(16,25,21,18);(6,7)(8,9)(10,11,14,17)(12,18,20,25)(13,21,23,16)(15,22,24,19);(3,4);(1,2)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10600470, #10600471, #10600474, …
maximal normal subgroup of	#10600470, #10600471, #10600474, …
maximal quotient of	C₂²×#3939, C₂²×#3942, C₂×#65259, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1481);