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C₂²×#674 #65742

order	320
isoclinism	isoclinic class of AGL(1,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

C₂²×#674

Splittings

C₅ ⋊ C₂²×C₁₆

Subgroups

	subgroup	G/H
center, hypercenter	C₂²×C₄	AGL(1,𝔽₅)
commutator, hypocenter	C₅	C₂²×C₁₆
Frattini	C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂²×C₄×C₅	C₄
socle, prime radical	C₂³×C₅	C₈
cosocle	#168	C₂³
LCS	G ▷ C₅
UCS	C₁ ◁ C₂²×C₄

98 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 25 points:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598698, #10598728, #10598777, …
maximal normal subgroup of	#10598698, #10598728, #10598777, …
maximal quotient of	C₂×C₄×#674, C₂×C₄⋊#674, #10604151, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1080);