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D660 #13050423

order1320
automorphismAut(G) order 105600
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD660
unambiguous semidirect productC4⋊D165

Splittings

FittingC3×C4×C5×C11 ⋊ C2
HallC3×C5×C11 ⋊ D4
HallC5×C11 ⋊ D12
HallC3×C11 ⋊ D20
HallC3×C5 ⋊ D44
SylowC11 ⋊ D60
SylowC5 ⋊ D132
SylowC3 ⋊ D220

Subgroups

subgroupG/H
center, FrattiniC2C2×D165
commutator, socle, cosocleC2×C3×C5×C11C22
FittingC3×C4×C5×C11C2
hypercenterC4D165
hypocenterC3×C5×C11D4
LCSG ▷ C2×C3×C5×C11 ▷ C3×C5×C11
UCSC1 ◁ C2 ◁ C4

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 660

A minimal faithful permutation representation — 2 generators on 23 points:

(1,2,3,4)(5,6,7)(8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23);(2,4)(6,7)(9,12)(10,11)(14,23)(15,22)(16,21)(17,20)(18,19)

Polycyclic presentation:

2^3,3,5,11|b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^10

Conjugacy classes & character table

333 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD1320, D1980, D3300, …
maximal normal subgroup ofD1320
maximal quotient ofD1320, Dic660, D1980, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1320, 127);