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C₂×#3993 #65719

order	320
isoclinism	isoclinic class of #3993
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂×#3993
unambiguous semidirect product	D₅⋊#106

Splittings

Fitting	C₂×C₅×C₈ ⋊ C₄
Sylow	C₅ ⋊ C₂×#106

Subgroups

	subgroup	G/H
center	C₂²	C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂²×C₄
Frattini	C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂×C₅×C₈	C₄
socle	C₂²×C₅	C₄⋊C₄
prime radical	C₂²×D₅	D₄
cosocle	C₄×D₅	C₂³
hypercenter	C₂×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#106
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

130 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)(8,9,10,12,11,13,14,15);(4,5,6,7)(9,12)(10,14)(13,15);(1,2)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598621, #10598627, #10598663, …
maximal normal subgroup of	#10598621, #10598627, #10598663, …
maximal quotient of	D₅⋊#334, #10603816, #10603817, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1057);