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C₂×C₂²⋊#170 #65797

order	320
isoclinism	isoclinic class of #700
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

C₂×C₂²⋊#170

Splittings

C₅ ⋊ C₂×C₂²⋊C₈

Subgroups

	subgroup	G/H
center	C₂³	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂²×C₈
Frattini	C₂²	C₂²×AGL(1,𝔽₅)
Fitting, prime radical	C₂⁴×C₅	C₄
socle	C₂³×C₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₂⁴	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₂²⋊C₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂⁴

202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598706, #10598821, #10604201, …
maximal normal subgroup of	#10598706, #10598821, #10604201, …
maximal quotient of	C₂×C₂²⋊#674, #10604872, C₂×#64912, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1135);