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C5×D3531 #47977226

order35310
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D3531
unambiguous semidirect productC3⋊(C5×D1177)
unambiguous semidirect productC107⋊(C5×D33)

Splittings

Fitting, socle, HallC3×C5×C11×C107 ⋊ C2
derived, HallC3×C11×C107 ⋊ C2×C5
HallC5×C11×C107 ⋊ D3
HallC3×C5×C107 ⋊ D11
HallC11×C107 ⋊ C5×D3
HallC5×C107 ⋊ D33
HallC3×C107 ⋊ C5×D11
HallC3×C5×C11 ⋊ D107
SylowC107 ⋊ C5×D33
HallC5×C11 ⋊ D321
HallC3×C11 ⋊ C5×D107
HallC3×C5 ⋊ D1177
SylowC11 ⋊ C5×D321
SylowC3 ⋊ C5×D1177

Subgroups

subgroupG/H
center, hypercenterC5D3531
commutator, cosocle, hypocenterC3×C11×C107C2×C5
FrattiniC1G
Fitting, socleC3×C5×C11×C107C2
LCSG ▷ C3×C11×C107
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 126
transitive degreeμt(G) = 17655

A minimal faithful permutation representation — 2 generators on 126 points:

(1,2,3,4,5)(6,7,8)(9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126);(7,8)(10,19)(11,18)(12,17)(13,16)(14,15)(21,126)(22,125)(23,124)(24,123)(25,122)(26,121)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,110)(38,109)(39,108)(40,107)(41,106)(42,105)(43,104)(44,103)(45,102)(46,101)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)

Conjugacy classes & character table

8835 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(35310, 13);