#64781

order	320
isoclinism	isoclinic class of #686
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #350

Subgroups

	subgroup	G/H
center	C₈	C₂²⋊D₅
commutator	C₄×C₅	C₂×C₈
Frattini	C₂×C₈	C₂×D₅
Fitting	C₅×#131	C₂
socle	C₂×C₅	C₂²⋊C₈
prime radical	C₅×#41	C₄
cosocle	C₂×C₅×C₈	C₂²
hypercenter	#131	D₅
hypocenter	C₅	#350
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #131

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 2 generators on 37 points:

(1,2,4,5,3)(6,8)(7,13)(9,18)(10,25)(11,19)(12,20)(14,27)(15,28)(16,29)(17,24)(21,35)(22,26)(23,33)(30,36)(31,37)(32,34);(2,3)(4,5)(6,7,9,14,11,15,22,30,12,16,23,31,10,24,21,34)(8,17,18,32,19,13,26,27,20,28,33,36,25,29,35,37)

Polycyclic presentation:

2^6,5|a^2=c,c^2=e,d^2=f,e^2=f,b^a=bd,d^a=df,g^a=g^4,d^b=df

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₅×#350, #10599918, #10601716, …
maximal normal subgroup of	D₅×#350, #10599918, #10601716, …
maximal quotient of	#10598162, #10598180, #10598209, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 119);