#65183

order	320
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#108 ⋊ C₂
Sylow	C₅ ⋊ #444

Subgroups

	subgroup	G/H
center	C₄	D₄×D₅
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#108	C₂
socle	C₂×C₅	C₄×D₄
prime radical	#4050	C₂
cosocle	C₂×C₄×C₅	C₂³
hypercenter	#108	D₅
hypocenter	C₅	#444
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

104 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)(6,9,10,13,12,17,11,21)(7,14,15,20,16,8,18,19);(6,8,11,20,12,14,10,19)(7,13,15,17,16,21,18,9);(2,3)(4,5)(6,7)(8,17)(9,14)(10,15)(11,18)(12,16)(13,20)(19,21)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10600047, #10600057, #10600102, …
maximal normal subgroup of	#10600047, #10600057, #10600102, …
maximal quotient of	#10599310, #10599311, #10599312, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 521);