#64735

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅×M₃₂ ⋊ C₂
Sylow	C₅ ⋊ #350

Subgroups

	subgroup	G/H
center	C₈	C₂²⋊D₅
commutator	C₄×C₅	C₂×C₈
Frattini	C₂×C₈	C₂×D₅
Fitting	C₅×M₃₂	C₂
socle	C₂×C₅	C₂²⋊C₈
prime radical	#709	C₄
cosocle	C₂×C₅×C₈	C₂²
hypercenter	M₃₂	D₅
hypocenter	C₅	#350
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

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

Polycyclic presentation:

2^6,5|b^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, #10600051, …
maximal normal subgroup of	D₅×#350, #10599918, #10600051, …
maximal quotient of	#10598006, #10598009, #10598014, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 73);