#65531

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #455

Subgroups

	subgroup	G/H
center	C₂	#4099
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting, prime radical	C₅×H(5,𝔽₂)	C₂
socle	C₂×C₅	#120
hypercenter	H(5,𝔽₂)	D₅
hypocenter	C₅	#455
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ H(5,𝔽₂)

152 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)(9,19)(12,17)(14,21)(16,18);(6,8)(7,11)(9,18)(10,15)(12,21)(13,20)(14,17)(16,19);(2,3)(4,5)(6,7,10,13)(8,14,15,21)(9,17,19,12)(11,18,20,16)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10598834, #10601790, #10601791, …
maximal normal subgroup of	#10598834, #10601790, #10601791, …
maximal quotient of	#10602440, #10602442, #10602458, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 869);