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C3⋊(C3⋊(C13⋊(D31⋊C5))) #49136606

order36270
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productC3⋊(C3⋊(C13⋊(D31⋊C5)))
unambiguous semidirect productC13⋊(C3⋊(C3⋊(D31⋊C5)))
unambiguous semidirect productΓL(1,𝔽32)⋊(C3⋊D39)
unambiguous semidirect product(C3⋊D1209)⋊C5

Splittings

Fitting, socle, derived, HallC32×C13×C31 ⋊ C2×C5
HallC32×C13×ΓL(1,𝔽32) ⋊ C2
HallC3⋊D1209 ⋊ C5
HallC13×ΓL(1,𝔽32) ⋊ C3⋊D3
HallC32×ΓL(1,𝔽32) ⋊ D13
HallC13×C31 ⋊ C5×C3⋊D3
HallC32×C31 ⋊ C5×D13
HallΓL(1,𝔽32) ⋊ C3⋊D39
HallC32×C13 ⋊ D31⋊C5
SylowC31 ⋊ C5×C3⋊D39
SylowC13 ⋊ C3⋊(C3⋊(D31⋊C5))
SylowC32 ⋊ C13⋊(D31⋊C5)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC32×C13×C31C2×C5
LCSG ▷ C32×C13×C31
UCSC1

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 50
transitive degreeμt(G) = 3627

A minimal faithful permutation representation — 3 generators on 50 points:

(2,3)(5,6)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(21,25,26,37,24,22,23,30,31,28)(27,42,43,44,29,33,34,46,48,36)(32,49,41,47,35,39,40,50,38,45);(4,5,6)(7,8,10,12,14,16,18,19,17,15,13,11,9)(20,21,24,29,31,38,48,42,26,32,41,49,43,45,33,25,23,27,35,46,40,50,39,30,34,44,47,37,36,28,22);(1,2,3)

Polycyclic presentation:

2,5,3^2,13,31|c^a=c^2,d^a=d^2,e^a=e^12,f^a=f^30,f^b=f^16

Conjugacy classes & character table

651 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36270, 126);