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C3⋊(C3⋊((C11⋊C5)⋊D31)) #44501832

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

Aliases

unambiguous semidirect productC3⋊(C3⋊((C11⋊C5)⋊D31))
unambiguous semidirect product(C11⋊C5)⋊(C3⋊D93)

Splittings

Fitting, socle, derived, HallC32×C11×C31 ⋊ C2×C5
HallC32×C31×C11⋊C5 ⋊ C2
HallC3⋊D1023 ⋊ C5
HallC31×C11⋊C5 ⋊ C3⋊D3
HallC32×C11⋊C5 ⋊ D31
HallC11×C31 ⋊ C5×C3⋊D3
HallC32×C31 ⋊ AGL(1,𝔽11)
HallC32×C11 ⋊ C5×D31
HallC11⋊C5 ⋊ C3⋊D93
SylowC31 ⋊ C3⋊(C3⋊AGL(1,𝔽11))
SylowC11 ⋊ C5×C3⋊D93
SylowC32 ⋊ (C11⋊C5)⋊D31

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

Conjugacy classes & character table

984 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(30690, 199);