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C5×C3⋊(#64408⋊C3) #27190261

order13140
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×C3⋊(#64408⋊C3)
unambiguous semidirect productC3⋊(C5×#64408⋊C3)
unambiguous semidirect product(C3⋊#64408)⋊(C3×C5)
unambiguous semidirect product(C5×C3⋊#64408)⋊C3

Splittings

Fitting, socleC3×C5×C73 ⋊ C3×C4
derivedC3×C73 ⋊ C3×C4×C5
HallC3×C5×C73⋊C3 ⋊ C4
HallC3×C73⋊C3 ⋊ C4×C5
HallC5×C73 ⋊ C3×Dic3
SylowC73 ⋊ C3×C5×Dic3

Subgroups

subgroupG/H
center, hypercenterC5C3⋊(#64408⋊C3)
commutator, hypocenterC3×C73C3×C4×C5
FrattiniC1G
Fitting, socleC3×C5×C73C3×C4
prime radicalC3×C5×D73⋊C3C2
cosocleC3×D73C2×C3×C5
LCSG ▷ C3×C73
UCSC1 ◁ C5

56 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:

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

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 95);