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C3⋊(C3⋊(C5⋊(D43⋊C7))) #41029145

order27090
automorphism?
homologyAbelianization (H1) = C2×C7 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productC3⋊(C3⋊(C5⋊(D43⋊C7)))
unambiguous semidirect productC5⋊(C3⋊(C3⋊(D43⋊C7)))
unambiguous semidirect product(C43⋊C7)⋊(C3⋊D15)
unambiguous semidirect product(C3⋊D645)⋊C7

Splittings

Fitting, socle, derived, HallC32×C5×C43 ⋊ C2×C7
HallC32×C5×C43⋊C7 ⋊ C2
HallC3⋊D645 ⋊ C7
HallC32×C43⋊C7 ⋊ D5
HallC5×C43⋊C7 ⋊ C3⋊D3
HallC32×C43 ⋊ C7×D5
HallC43⋊C7 ⋊ C3⋊D15
HallC5×C43 ⋊ C7×C3⋊D3
HallC32×C5 ⋊ D43⋊C7
SylowC43 ⋊ C7×C3⋊D15
SylowC32 ⋊ C5⋊(D43⋊C7)
SylowC5 ⋊ C3⋊(C3⋊(D43⋊C7))

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC32×C5×C43C2×C7
LCSG ▷ C32×C5×C43
UCSC1

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 54
transitive degreeμt(G) = 1935

A minimal faithful permutation representation — 3 generators on 54 points:

(2,5,6,11,14,30,35,3,4,7,9,19,22,42)(8,18,21,41,10,26,27,12,13,29,33,16,17,40)(15,38,39,43,23,28,32,24,25,31,34,36,20,37)(45,46)(47,48)(50,51)(53,54);(1,2,5,8,6,10,18,24,11,20,26,22,21,34,25,39,14,23,37,33,27,35,42,40,41,32,36,19,31,38,43,29,30,17,28,9,15,13,16,7,12,4,3)(44,45,47,48,46)(52,53,54);(49,50,51)

Polycyclic presentation:

2,7,3^2,5,43|c^a=c^2,d^a=d^2,e^a=e^4,f^a=f^42,f^b=f^41

Conjugacy classes & character table

303 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(27090, 86);