#65402

order	320
isoclinism	isoclinic class of #677
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₂²×C₅×C₈ ⋊ C₂
Sylow	C₅ ⋊ #416

Subgroups

	subgroup	G/H
center	C₂×C₄	D₂₀
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₂²×C₅×C₈	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂×#709	C₂
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₂²×C₈	D₅
hypocenter	C₅	#416
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 25 points:

(1,2,4,5,3)(6,7,8,9)(10,13,12,19,15,22,14,21)(11,17,23,25,18,24,16,20);(10,12,15,14)(11,16,18,23)(13,19,22,21)(17,20,24,25);(2,3)(4,5)(10,11)(12,16)(13,20)(14,23)(15,18)(17,21)(19,24)(22,25)

Polycyclic presentation:

2^6,5|b^2=f,c^2=de,d^2=f,c^a=cd,d^a=df,g^a=g^4

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10599439, #10599487, #10599711, …
maximal normal subgroup of	#10599439, #10599487, #10599711, …
maximal quotient of	#10600527, #10600529, C₄×#3953, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 740);