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C59⋊(D37⋊C3) #27189883

order13098
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC59⋊(D37⋊C3)
unambiguous semidirect product(C37⋊C3)⋊D59
unambiguous semidirect productD2183⋊C3

Splittings

Fitting, socle, derived, HallC37×C59 ⋊ C2×C3
HallC59×C37⋊C3 ⋊ C2
HallD2183 ⋊ C3
HallC37⋊C3 ⋊ D59
SylowC59 ⋊ D37⋊C3
SylowC37 ⋊ C3×D59

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC37×C59C2×C3
LCSG ▷ C37×C59
UCSC1

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 96
transitive degreeμt(G) = 2183

A minimal faithful permutation representation — 2 generators on 96 points:

(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)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(61,65,66,62,63,67)(64,73,74,68,69,82)(70,85,86,76,77,93)(71,84,80,78,75,72)(79,95,96,88,87,91)(81,83,94,90,89,92);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(60,61,64,70,77,87,92,93,90,80,66,65,71,79,89,69,75,82,91,96,74,84,73,83,88,78,63,67,72,81,86,94,95,85,76,68,62)

Polycyclic presentation:

2,3,37,59|c^a=c^36,d^a=d^58,c^b=c^26

Conjugacy classes & character table

447 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD59×D37⋊C3, C2×C59⋊(D37⋊C3)
maximal normal subgroup ofD59×D37⋊C3, C2×C59⋊(D37⋊C3)
maximal quotient ofC59⋊(Dic37⋊C3), C2×C59⋊(D37⋊C3)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13098, 12);