#65291

order	320
isoclinism	isoclinic class of #686
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#117 ⋊ C₂
Sylow	C₅ ⋊ #416

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₅
commutator	C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#117	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂×#709	C₂
cosocle	C₂×C₄×C₅	C₂³
hypercenter	#117	D₅
hypocenter	C₅	#416
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 25 points:

(1,2,4,5,3)(6,7,8,9)(11,16)(13,21)(15,22)(17,18)(19,25)(23,24);(10,12,14,20)(11,15,18,24)(13,19,21,25)(16,22,17,23);(2,3)(4,5)(10,11)(12,15)(13,16)(14,18)(17,21)(19,22)(20,24)(23,25)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10599491, #10599493, #10599532, …
maximal normal subgroup of	#10599491, #10599493, #10599532, …
maximal quotient of	#10600224, #10600236, C₄×#3941, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 629);