#65167

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

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #499

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	#499
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

86 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)(6,8,9,13)(7,10,11,12)(15,20)(16,22)(17,25)(19,26)(23,24)(27,29);(14,16,17,23,18,24,25,22)(15,19,21,26,20,27,28,29);(2,3)(4,5)(6,7,9,11)(8,12,13,10)(14,15)(16,19)(17,21)(18,20)(22,29)(23,26)(24,27)(25,28)

Polycyclic presentation:

2^6,5|a^2=d,b^2=e,c^2=d,e^2=f,c^a=cdf,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	#10598615, #10598616, #10599973, …
maximal normal subgroup of	#10598615, #10598616, #10599973, …
maximal quotient of	#10599597, #10599622, #10599630, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 505);