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C3⋊(C3⋊(C7⋊(D41⋊C5))) #38788951

order25830
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C3

Aliases

unambiguous semidirect productC3⋊(C3⋊(C7⋊(D41⋊C5)))
unambiguous semidirect productC7⋊(C3⋊(C3⋊(D41⋊C5)))
unambiguous semidirect product(C41⋊C5)⋊(C3⋊D21)
unambiguous semidirect product(C3⋊D861)⋊C5

Splittings

Fitting, socle, derived, HallC32×C7×C41 ⋊ C2×C5
HallC32×C7×C41⋊C5 ⋊ C2
HallC3⋊D861 ⋊ C5
HallC32×C41⋊C5 ⋊ D7
HallC7×C41⋊C5 ⋊ C3⋊D3
HallC32×C41 ⋊ C5×D7
HallC7×C41 ⋊ C5×C3⋊D3
HallC41⋊C5 ⋊ C3⋊D21
HallC32×C7 ⋊ D41⋊C5
SylowC41 ⋊ C5×C3⋊D21
SylowC32 ⋊ C7⋊(D41⋊C5)
SylowC7 ⋊ C3⋊(C3⋊(D41⋊C5))

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC32×C7×C41C2×C5
LCSG ▷ C32×C7×C41
UCSC1

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 54
transitive degreeμt(G) = 2583

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

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

Polycyclic presentation:

2,5,3^2,7,41|c^a=c^2,d^a=d^2,e^a=e^6,f^a=f^40,f^b=f^10

Conjugacy classes & character table

417 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25830, 55);