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C3⋊(C3⋊#16811785) #44501826

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

Aliases

unambiguous semidirect productC3⋊(C3⋊#16811785)

Splittings

Fitting, socle, derived, HallC32×C11×C31 ⋊ C2×C5
HallC32×#13133722 ⋊ C2
HallC3⋊D1023 ⋊ C5
Hall#13133722 ⋊ C3⋊D3
HallC11×C31 ⋊ C5×C3⋊D3
HallC32×C31 ⋊ AGL(1,𝔽11)
HallC32×C11 ⋊ D31⋊C5
SylowC31 ⋊ C3⋊(C3⋊AGL(1,𝔽11))
SylowC11 ⋊ C3⋊(C3⋊(D31⋊C5))
SylowC32 ⋊ #16811785

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

Conjugacy classes & character table

336 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30690, 193);