#65207

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

Aliases

No aliases for this group.

Splittings

Fitting	C₅×Q₃₂ ⋊ C₂
Sylow	C₅ ⋊ #510

Subgroups

	subgroup	G/H
center	C₂	D₅×D₈
commutator, cosocle	C₅×C₈	C₂³
Frattini	C₈	C₂²×D₅
Fitting	C₅×Q₃₂	C₂
socle	C₂×C₅	C₂×D₈
prime radical	#4065	C₂
hypercenter	Q₃₂	D₅
hypocenter	C₅	#510
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

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

Polycyclic presentation:

2^6,5|b^2=f,c^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	#10604138, #10604139, C₂×#65207, …
maximal normal subgroup of	#10604138, #10604139, C₂×#65207, …
maximal quotient of	#10600001, #10600002, #10600003, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 545);