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C3×#13091733⋊C3 #27190256

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

Aliases

direct productC3×#13091733⋊C3
unambiguous semidirect product#13091733⋊C32
unambiguous semidirect product(C3×#13091733)⋊C3

Splittings

Fitting, socleC3×C5×C73 ⋊ C3×C4
derived, HallC5×C73 ⋊ C32×C4
HallC3×C5×C73⋊C3 ⋊ C4
Hall#13091733 ⋊ C32
HallC3×C73⋊C3 ⋊ AGL(1,𝔽5)
SylowC73 ⋊ C32×AGL(1,𝔽5)
SylowC5 ⋊ C3×#64408⋊C3

Subgroups

subgroupG/H
center, hypercenterC3#13091733⋊C3
commutator, hypocenterC5×C73C32×C4
FrattiniC1G
Fitting, socleC3×C5×C73C3×C4
prime radicalC3×C5⋊(D73⋊C3)C2
cosocleD365C2×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:

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

Conjugacy classes & character table

135 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 90);