#65166

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #479

Subgroups

	subgroup	G/H
center	C₂²	D₄×D₅
commutator, cosocle	C₂×C₄×C₅	C₂³
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₅×#107	C₂
socle, prime radical	C₂²×C₅	C₂×D₄
hypercenter	#107	D₅
hypocenter	C₅	#479
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 320

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10610886, #10610890, #10610893, …
maximal normal subgroup of	#10610886, #10610890, #10610893, …
maximal quotient of	#10599582, #10599595, #10599602, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 504);