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C3⋊(C3⋊(ΓL(1,𝔽32)⋊D11)) #44501843

order30690
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productC3⋊(C3⋊(ΓL(1,𝔽32)⋊D11))
unambiguous semidirect productΓL(1,𝔽32)⋊(C3⋊D33)

Splittings

Fitting, socle, derived, HallC32×C11×C31 ⋊ C2×C5
HallC32×C11×ΓL(1,𝔽32) ⋊ C2
HallC3⋊D1023 ⋊ C5
HallC11×ΓL(1,𝔽32) ⋊ C3⋊D3
HallC32×ΓL(1,𝔽32) ⋊ D11
HallC11×C31 ⋊ C5×C3⋊D3
HallC32×C31 ⋊ C5×D11
HallΓL(1,𝔽32) ⋊ C3⋊D33
HallC32×C11 ⋊ D31⋊C5
SylowC31 ⋊ C5×C3⋊D33
SylowC11 ⋊ C3⋊(C3⋊(D31⋊C5))
SylowC32 ⋊ ΓL(1,𝔽32)⋊D11

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC32×C11×C31C2×C5
LCSG ▷ C32×C11×C31
UCSC1

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 48
transitive degreeμt(G) = 3069

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2,5,3^2,11,31|c^a=c^2,d^a=d^2,e^a=e^10,f^a=f^30,f^b=f^16

Conjugacy classes & character table

552 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30690, 210);