#65532

order	320
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #458

Subgroups

	subgroup	G/H
center	C₂	#4099
commutator, cosocle	C₂³×C₅	C₂³
Frattini	C₂³	C₂²×D₅
Fitting, prime radical	C₅×H(5,𝔽₂)	C₂
socle	C₂×C₅	#120
hypercenter	H(5,𝔽₂)	D₅
hypocenter	C₅	#458
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ H(5,𝔽₂)

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 21 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10601778, #10601779, #10601783, …
maximal normal subgroup of	#10601778, #10601779, #10601783, …
maximal quotient of	#10602448, #10602450, #10602636, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 870);