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C₂×#4004 #65752

order	320
isoclinism	isoclinic class of #700
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

Splittings

C₅ ⋊ C₂×C₄⋊C₈

Subgroups

	subgroup	G/H
center	C₂³	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂²×C₈
Frattini	C₂²	C₂²×AGL(1,𝔽₅)
Fitting	C₂²×C₄×C₅	C₄
socle, prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₂²×C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 3 generators on 19 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598562, #10598563, #10598725, …
maximal normal subgroup of	#10598562, #10598563, #10598725, …
maximal quotient of	C₂×#64861, C₄×#4004, #10604187, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1090);