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C₂×#3992 #65717

order	320
isoclinism	isoclinic class of #700
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

Splittings

Fitting	C₂×C₅×C₈ ⋊ C₄
Sylow	C₅ ⋊ C₂×#97

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄²
Frattini	C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×C₈	C₄
socle	C₂²×C₅	C₄²
prime radical	C₂²×D₅	C₂×C₄
cosocle	C₄×D₅	C₂³
hypercenter	C₂×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#97
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 15 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598660, #10598671, C₄×#3992, …
maximal normal subgroup of	#10598660, #10598671, C₄×#3992, …
maximal quotient of	#10603806, #10603807, #10603810, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1055);