#65177

order	320
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting	C₅×#107 ⋊ C₂
Sylow	C₅ ⋊ #485

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	#485
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 29 points:

(1,2,4,5,3)(7,11)(10,13)(14,16,23,29,18,24,17,22)(15,20,26,27,19,25,21,28);(6,8,9,12)(7,10,11,13)(15,19)(16,22)(17,23)(20,27)(24,29)(25,28);(2,3)(4,5)(6,7)(8,10)(9,11)(12,13)(14,15)(16,20)(17,21)(18,19)(22,28)(23,26)(24,25)(27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10610872, C₂×#65177, #10610886, …
maximal normal subgroup of	#10610872, C₂×#65177, #10610886, …
maximal quotient of	#10599586, #10599610, #10599617, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 515);