#65436

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

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₅×D₈ ⋊ C₂
Sylow	C₅ ⋊ #509

Subgroups

	subgroup	G/H
center	C₂	C₂×#686
commutator, cosocle	C₅×C₈	C₂³
Frattini	C₈	C₂²×D₅
Fitting	C₂×C₅×D₈	C₂
socle	C₂×C₅	C₂×D₈
hypercenter	C₂×D₈	D₅
hypocenter	C₅	#509
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₈

90 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 3 generators on 21 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10600047, #10600055, #10600056, …
maximal normal subgroup of	#10600047, #10600055, #10600056, …
maximal quotient of	#10601389, #10601391, #10601393, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 774);