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C2×C3⋊(C3⋊(D101⋊C5)) #32911450

order18180
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3⋊(C3⋊(D101⋊C5))
unambiguous semidirect product(C2×C3)⋊(C3⋊(D101⋊C5))
unambiguous semidirect product(C2×C101⋊C5)⋊(C3⋊D3)
unambiguous semidirect product(C2×C3⋊D303)⋊C5

Splittings

Fitting, socleC2×C32×C101 ⋊ C2×C5
derived, HallC32×C101 ⋊ C22×C5
HallC32×C101⋊C5 ⋊ C22
HallC2×C3⋊D303 ⋊ C5
HallC101⋊C5 ⋊ C2×C3⋊D3
SylowC101 ⋊ C2×C5×C3⋊D3
SylowC32 ⋊ C2×D101⋊C5

Subgroups

subgroupG/H
center, hypercenterC2C3⋊(C3⋊(D101⋊C5))
commutator, cosocle, hypocenterC32×C101C22×C5
FrattiniC1G
Fitting, socleC2×C32×C101C2×C5
LCSG ▷ C32×C101
UCSC1 ◁ C2

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 109
transitive degreeμt(G) = 1818

A minimal faithful permutation representation — 3 generators on 109 points:

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

Conjugacy classes & character table

240 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18180, 83);