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C3×C3⋊#13091732 #27190293

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

Aliases

direct productC3×C3⋊#13091732
unambiguous semidirect productC3⋊(C3×#13091732)

Splittings

Fitting, socle, HallC32×C5×C73 ⋊ C4
derivedC3×C5×C73 ⋊ C3×C4
HallC32×C73 ⋊ AGL(1,𝔽5)
HallC5×C73 ⋊ C3×Dic3
SylowC73 ⋊ C3×ΓL(1,𝔽16)
HallC32×C5 ⋊ #64408
SylowC32 ⋊ #13091732
SylowC5 ⋊ C3×C3⋊#64408

Subgroups

subgroupG/H
center, hypercenterC3C3⋊#13091732
commutator, hypocenterC3×C5×C73C3×C4
FrattiniC1G
Fitting, socleC32×C5×C73C4
prime radicalC32×D365C2
cosocleC3×D365C2×C3
LCSG ▷ C3×C5×C73
UCSC1 ◁ C3

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 84
transitive degreeμt(G) = 2190

A minimal faithful permutation representation — 2 generators on 84 points:

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

Conjugacy classes & character table

837 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 127);