#65204

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅×SD₃₂ ⋊ C₂
Sylow	C₅ ⋊ #510

Subgroups

	subgroup	G/H
center	C₂	D₅×D₈
commutator, cosocle	C₅×C₈	C₂³
Frattini	C₈	C₂²×D₅
Fitting	C₅×SD₃₂	C₂
socle	C₂×C₅	C₂×D₈
prime radical	#4058	C₂
hypercenter	SD₃₂	D₅
hypocenter	C₅	#510
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

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

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10604130, #10604131, C₂×#65204, …
maximal normal subgroup of	#10604130, #10604131, C₂×#65204, …
maximal quotient of	#10599969, #10599971, #10599978, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 542);