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C₂²×#700 #66250

order	320
isoclinism	isoclinic class of #700
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

C₂²×#700

Splittings

C₅ ⋊ C₂²×M₁₆

Subgroups

	subgroup	G/H
center	C₂³	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂³×C₄
Frattini	C₂	C₂³×AGL(1,𝔽₅)
Fitting	C₂²×C₄×C₅	C₄
socle	C₂³×C₅	C₂×C₄
prime radical	C₂³×D₅	C₄
cosocle	Dic₅	C₂⁴
hypercenter	C₂²×C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₂²×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 17 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10603777, #10603786, #10603787, …
maximal normal subgroup of	#10603777, #10603786, #10603787, …
maximal quotient of	C₂×C₄×#700, C₂×#65681, C₂×#65680, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1588);