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C3×C3⋊#13091734 #27190295

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

Aliases

direct productC3×C3⋊#13091734
unambiguous semidirect productC3⋊(C3×#13091734)

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 ⋊ Dic73
SylowC32 ⋊ #13091734
SylowC5 ⋊ C3×Dic219

Subgroups

subgroupG/H
center, hypercenterC3C3⋊#13091734
commutator, hypocenterC3×C5×C73C3×C4
FrattiniC1G
Fitting, socleC32×C5×C73C4
prime radicalC32×D5×C73C2
cosocleC3×D5×C73C2×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:

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

Conjugacy classes & character table

1323 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13140, 129);