#65176

order	320
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#107 ⋊ C₂
Sylow	C₅ ⋊ #467

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#107	C₂
socle	C₂²×C₅	C₂×D₄
prime radical	C₂²×D₅	D₄
hypercenter	#107	D₅
hypocenter	C₅	#467
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

114 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)(12,18)(14,21)(15,22)(19,20)(23,25);(10,12,14,20,13,19,21,18)(11,15,17,22,16,23,24,25);(2,3)(4,5)(7,9)(10,11)(12,15)(13,16)(14,17)(18,25)(19,23)(20,22)(21,24)

Polycyclic presentation:

2^6,5|b^2=e,c^2=df,e^2=f,c^a=cd,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	#10598612, #10598614, #10599986, …
maximal normal subgroup of	#10598612, #10598614, #10599986, …
maximal quotient of	#10599370, #10599372, #10599374, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 514);