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C3×D5×D73⋊C3 #27190260

order13140
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3

Aliases

direct productC3×D5×D73⋊C3
unambiguous semidirect productD73⋊(C32×D5)
unambiguous semidirect product(D5×D73)⋊C32
unambiguous semidirect product(C5×D73⋊C3)⋊(C2×C3)
unambiguous semidirect product(D5×C73⋊C3)⋊(C2×C3)
unambiguous semidirect product(C5⋊(D73⋊C3))⋊(C2×C3)
unambiguous semidirect product(C3×D5×D73)⋊C3

Splittings

Fitting, socleC3×C5×C73 ⋊ C22×C3
derived, HallC5×C73 ⋊ C22×C32
HallC3×C5×C73⋊C3 ⋊ C22
HallD5×D73 ⋊ C32
HallC3×C73⋊C3 ⋊ C2×D5
SylowC73 ⋊ C2×C32×D5
SylowC5 ⋊ C2×C3×D73⋊C3

Subgroups

subgroupG/H
center, hypercenterC3D5×D73⋊C3
commutator, cosocle, hypocenterC5×C73C22×C32
FrattiniC1G
Fitting, socleC3×C5×C73C22×C3
LCSG ▷ C5×C73
UCSC1 ◁ C3

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 81
transitive degreeμt(G) = 1095

A minimal faithful permutation representation — 2 generators on 81 points:

(2,6,7,3,4,8)(5,14,15,9,10,24)(11,27,28,17,18,41)(12,26,22,20,16,13)(19,46,47,30,31,60)(21,44,45,33,29,38)(23,25,43,35,36,39)(32,48,57,34,49,63)(37,42,62,50,51,58)(40,67,65,54,55,53)(52,61,68,56,64,72)(59,73,71,69,70,66)(74,75,77,78,76);(1,2,5,11,19,32,33,20,4,8,13,23,37,52,54,36,10,16,24,38,53,66,69,51,18,29,39,41,57,68,73,64,31,48,55,58,60,47,62,67,49,46,61,70,72,63,28,43,44,27,42,59,71,65,45,15,26,14,25,40,56,50,35,22,7,6,12,21,34,30,17,9,3)(75,76)(77,78)(79,80,81)

Conjugacy classes & character table

216 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 94);