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C3×D3535 #35234584

order21210
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productC3×D3535
unambiguous semidirect productC5⋊(C3×D707)
unambiguous semidirect productC101⋊(C3×D35)

Splittings

Fitting, socle, HallC3×C5×C7×C101 ⋊ C2
derived, HallC5×C7×C101 ⋊ C2×C3
HallC3×C7×C101 ⋊ D5
HallC3×C5×C101 ⋊ D7
HallC7×C101 ⋊ C3×D5
HallC5×C101 ⋊ C3×D7
HallC3×C101 ⋊ D35
HallC3×C5×C7 ⋊ D101
SylowC101 ⋊ C3×D35
HallC5×C7 ⋊ C3×D101
HallC3×C7 ⋊ D505
HallC3×C5 ⋊ D707
SylowC7 ⋊ C3×D505
SylowC5 ⋊ C3×D707

Subgroups

subgroupG/H
center, hypercenterC3D3535
commutator, cosocle, hypocenterC5×C7×C101C2×C3
FrattiniC1G
Fitting, socleC3×C5×C7×C101C2
LCSG ▷ C5×C7×C101
UCSC1 ◁ C3

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 116
transitive degreeμt(G) = 10605

A minimal faithful permutation representation — 2 generators on 116 points:

(1,2,3)(4,5,6,7,8)(9,10,11,12,13,14,15)(16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116);(5,8)(6,7)(10,15)(11,14)(12,13)(17,116)(18,115)(19,114)(20,113)(21,112)(22,111)(23,110)(24,109)(25,108)(26,107)(27,106)(28,105)(29,104)(30,103)(31,102)(32,101)(33,100)(34,99)(35,98)(36,97)(37,96)(38,95)(39,94)(40,93)(41,92)(42,91)(43,90)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)

Conjugacy classes & character table

5307 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21210, 11);