#64728

order	320
isoclinism	isoclinic class of #677
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅×C₁₆ ⋊ C₂
Sylow	C₅ ⋊ #350

Subgroups

	subgroup	G/H
center	C₈	D₂₀
commutator	C₄×C₅	C₂×C₈
Frattini	C₂×C₈	C₂×D₅
Fitting	C₂×C₅×C₁₆	C₂
socle	C₂×C₅	C₂²⋊C₈
prime radical	#709	C₄
cosocle	C₂×C₅×C₈	C₂²
hypercenter	C₂×C₁₆	D₅
hypocenter	C₅	#350
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂×C₁₆

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10599849, #10599869, #10599915, …
maximal normal subgroup of	#10599849, #10599869, #10599915, …
maximal quotient of	#10597994, #10597997, #10598014, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 66);