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C3×#13091735⋊C3 #27190259

order13140
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C3

Aliases

direct productC3×#13091735⋊C3
unambiguous semidirect product#13091735⋊C32
unambiguous semidirect product(C3×#13091735)⋊C3

Splittings

Fitting, socleC3×C5×C73 ⋊ C3×C4
derived, HallC5×C73 ⋊ C32×C4
HallC3×C5×C73⋊C3 ⋊ C4
Hall#13091735 ⋊ C32
HallC3×C73⋊C3 ⋊ Dic5
SylowC73 ⋊ C32×Dic5
SylowC5 ⋊ C3×#64408⋊C3

Subgroups

subgroupG/H
center, hypercenterC3#13091735⋊C3
commutator, hypocenterC5×C73C32×C4
FrattiniC1G
Fitting, socleC3×C5×C73C3×C4
prime radicalC3×C5×D73⋊C3C2
cosocleC5×D73C2×C32
LCSG ▷ C5×C73
UCSC1 ◁ C3

72 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:

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

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 93);