Finite Group Databasebeta

← Prev (#41004213) Next (#41004215) →

C5⋊(C29⋊(D31⋊C3)) #41004214

order26970
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊(C29⋊(D31⋊C3))
unambiguous semidirect productC29⋊(C5⋊(D31⋊C3))
unambiguous semidirect product(C31⋊C3)⋊D145
unambiguous semidirect productD4495⋊C3

Splittings

Fitting, socle, derived, HallC5×C29×C31 ⋊ C2×C3
HallC5×C29×C31⋊C3 ⋊ C2
HallD4495 ⋊ C3
HallC29×C31⋊C3 ⋊ D5
HallC29×C31 ⋊ C3×D5
HallC5×C31⋊C3 ⋊ D29
HallC5×C31 ⋊ C3×D29
HallC5×C29 ⋊ D31⋊C3
HallC31⋊C3 ⋊ D145
SylowC31 ⋊ C3×D145
SylowC29 ⋊ C5⋊(D31⋊C3)
SylowC5 ⋊ C29⋊(D31⋊C3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C29×C31C2×C3
LCSG ▷ C5×C29×C31
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 65
transitive degreeμt(G) = 4495

A minimal faithful permutation representation — 2 generators on 65 points:

(2,6,7,3,4,8)(5,14,15,9,10,22)(11,24,25,17,18,30)(12,23,19,20,16,13)(21,31,28,27,26,29)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60)(62,63)(64,65);(1,2,5,11,19,7,6,12,21,28,15,23,14,18,26,25,30,31,24,10,16,22,29,27,20,4,8,13,17,9,3)(32,33,35,37,39,41,43,45,47,49,51,53,55,57,59,60,58,56,54,52,50,48,46,44,42,40,38,36,34)(61,62,64,65,63)

Polycyclic presentation:

2,3,5,29,31|c^a=c^4,d^a=d^28,e^a=e^30,e^b=e^25

Conjugacy classes & character table

947 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26970, 33);