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C5×D3333 #45686786

order33330
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×D3333
unambiguous semidirect productC3⋊(C5×D1111)

Splittings

Fitting, socle, HallC3×C5×C11×C101 ⋊ C2
derived, HallC3×C11×C101 ⋊ C2×C5
HallC5×C11×C101 ⋊ D3
HallC3×C5×C101 ⋊ D11
HallC11×C101 ⋊ C5×D3
HallC5×C101 ⋊ D33
HallC3×C101 ⋊ C5×D11
HallC3×C5×C11 ⋊ D101
SylowC101 ⋊ C5×D33
HallC5×C11 ⋊ D303
HallC3×C11 ⋊ C5×D101
HallC3×C5 ⋊ D1111
SylowC11 ⋊ C5×D303
SylowC3 ⋊ C5×D1111

Subgroups

subgroupG/H
center, hypercenterC5D3333
commutator, cosocle, hypocenterC3×C11×C101C2×C5
FrattiniC1G
Fitting, socleC3×C5×C11×C101C2
LCSG ▷ C3×C11×C101
UCSC1 ◁ C5

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 120
transitive degreeμt(G) = 16665

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

Conjugacy classes & character table

8340 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33330, 18);