#65198

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅×M₃₂ ⋊ C₂
Sylow	C₅ ⋊ #510

Subgroups

	subgroup	G/H
center	C₂	C₂×D₄₀
commutator, cosocle	C₅×C₈	C₂³
Frattini	C₈	C₂²×D₅
Fitting	C₅×M₃₂	C₂
socle	C₂×C₅	C₂×D₈
prime radical	#4050	C₂
hypercenter	M₃₂	D₅
hypocenter	C₅	#510
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

82 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 37
transitive degree	μ_t(G) = 160

A minimal faithful permutation representation — 3 generators on 37 points:

(1,2,4,5,3)(6,9,18,33,19,21,10,22,11,23,30,36,8,17,25,37)(7,13,28,34,12,26,14,20,16,29,24,32,15,27,31,35);(6,8,11,19)(7,12,16,15)(9,21,23,17)(10,18,25,30)(13,27,29,26)(14,24,31,28)(20,34,35,32)(22,36,37,33);(2,3)(4,5)(6,7)(8,12)(9,20)(10,24)(11,16)(13,22)(14,30)(15,19)(17,32)(18,31)(21,34)(23,35)(25,28)(26,33)(27,36)(29,37)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10599883, #10599914, #10599921, …
maximal normal subgroup of	#10599883, #10599914, #10599921, …
maximal quotient of	#10599862, #10599870, #10599872, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 536);