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C5⋊(C29⋊(D37⋊C3)) #44565731

order32190
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊(C29⋊(D37⋊C3))
unambiguous semidirect productC29⋊(C5⋊(D37⋊C3))
unambiguous semidirect product(C37⋊C3)⋊D145
unambiguous semidirect productD5365⋊C3

Splittings

Fitting, socle, derived, HallC5×C29×C37 ⋊ C2×C3
HallC5×C29×C37⋊C3 ⋊ C2
HallD5365 ⋊ C3
HallC29×C37⋊C3 ⋊ D5
HallC29×C37 ⋊ C3×D5
HallC5×C37⋊C3 ⋊ D29
HallC5×C37 ⋊ C3×D29
HallC5×C29 ⋊ D37⋊C3
HallC37⋊C3 ⋊ D145
SylowC37 ⋊ C3×D145
SylowC29 ⋊ C5⋊(D37⋊C3)
SylowC5 ⋊ C29⋊(D37⋊C3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C29×C37C2×C3
LCSG ▷ C5×C29×C37
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 71
transitive degreeμt(G) = 5365

A minimal faithful permutation representation — 2 generators on 71 points:

(2,3)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(36,40,41,37,38,42)(39,48,49,43,44,57)(45,60,61,51,52,68)(46,59,55,53,50,47)(54,70,71,63,62,66)(56,58,69,65,64,67);(1,2,4,5,3)(6,7,9,11,13,15,17,19,21,23,25,27,29,31,33,34,32,30,28,26,24,22,20,18,16,14,12,10,8)(35,36,39,45,52,62,67,68,65,55,41,40,46,54,64,44,50,57,66,71,49,59,48,58,63,53,38,42,47,56,61,69,70,60,51,43,37)

Polycyclic presentation:

2,3,5,29,37|c^a=c^4,d^a=d^28,e^a=e^36,e^b=e^26

Conjugacy classes & character table

1092 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32190, 24);