#64737

order	320
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Sylow

C₅ ⋊ #362

Subgroups

	subgroup	G/H
center	C₂	#3953
commutator	C₅×C₈	C₂×C₄
Frattini	C₂×C₈	C₂×D₅
Fitting	C₅×M₃₂	C₂
socle	C₂×C₅	#102
prime radical	C₂²×C₅	SD₁₆
cosocle	C₂×C₅×C₈	C₂²
hypercenter	M₃₂	D₅
hypocenter	C₅	#362
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 37 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10599883, #10599910, #10599920, …
maximal normal subgroup of	#10599883, #10599910, #10599920, …
maximal quotient of	#10598008, #10598020, #10598052, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 75);