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D760 #13095083

order1520
automorphismAut(G) order 218880
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD760

Splittings

FittingC5×C8×C19 ⋊ C2
HallC5×C19 ⋊ D8
SylowC19 ⋊ D40
SylowC5 ⋊ D152

Subgroups

subgroupG/H
centerC2D380
commutator, cosocleC4×C5×C19C22
FrattiniC4C2×D95
FittingC5×C8×C19C2
socleC2×C5×C19D4
hypercenterC8D95
hypocenterC5×C19D8
LCSG ▷ C4×C5×C19 ▷ C2×C5×C19 ▷ C5×C19
UCSC1 ◁ C2 ◁ C4 ◁ C8

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 32
transitive degreeμt(G) = 760

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^4,5,19|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^4,f^a=f^18

Conjugacy classes & character table

383 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD1520, D2280, D3800, …
maximal normal subgroup ofD1520
maximal quotient ofD1520, D2280, Dic760, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1520, 67);