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C3⋊(C3⋊(C7⋊(D31⋊C5))) #34086221

order19530
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productC3⋊(C3⋊(C7⋊(D31⋊C5)))
unambiguous semidirect productC7⋊(C3⋊(C3⋊(D31⋊C5)))
unambiguous semidirect productΓL(1,𝔽32)⋊(C3⋊D21)
unambiguous semidirect product(C3⋊D651)⋊C5

Splittings

Fitting, socle, derived, HallC32×C7×C31 ⋊ C2×C5
HallC32×C7×ΓL(1,𝔽32) ⋊ C2
HallC3⋊D651 ⋊ C5
HallC32×ΓL(1,𝔽32) ⋊ D7
HallC7×ΓL(1,𝔽32) ⋊ C3⋊D3
HallC32×C31 ⋊ C5×D7
HallC7×C31 ⋊ C5×C3⋊D3
HallΓL(1,𝔽32) ⋊ C3⋊D21
HallC32×C7 ⋊ D31⋊C5
SylowC31 ⋊ C5×C3⋊D21
SylowC32 ⋊ C7⋊(D31⋊C5)
SylowC7 ⋊ C3⋊(C3⋊(D31⋊C5))

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC32×C7×C31C2×C5
LCSG ▷ C32×C7×C31
UCSC1

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 44
transitive degreeμt(G) = 1953

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2,5,3^2,7,31|c^a=c^2,d^a=d^2,e^a=e^6,f^a=f^30,f^b=f^16

Conjugacy classes & character table

354 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19530, 126);