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C₂×#4003 #65751

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

Aliases

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	C₂²×C₄×C₅	C₄
socle	C₂³×C₅	C₂×C₄
prime radical	C₂³×D₅	C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₂²×C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₂×C₂²⋊C₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598559, #10598561, #10598720, …
maximal normal subgroup of	#10598559, #10598561, #10598720, …
maximal quotient of	#10604159, C₄×#4003, #10604182, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1089);