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C5⋊(C17⋊(D61⋊C3)) #44527460

order31110
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC5⋊(C17⋊(D61⋊C3))
unambiguous semidirect productC17⋊(C5⋊(D61⋊C3))
unambiguous semidirect product(C61⋊C3)⋊D85
unambiguous semidirect productD5185⋊C3

Splittings

Fitting, socle, derived, HallC5×C17×C61 ⋊ C2×C3
HallC5×C17×C61⋊C3 ⋊ C2
HallD5185 ⋊ C3
HallC17×C61⋊C3 ⋊ D5
HallC17×C61 ⋊ C3×D5
HallC5×C61⋊C3 ⋊ D17
HallC5×C61 ⋊ C3×D17
HallC61⋊C3 ⋊ D85
HallC5×C17 ⋊ D61⋊C3
SylowC61 ⋊ C3×D85
SylowC17 ⋊ C5⋊(D61⋊C3)
SylowC5 ⋊ C17⋊(D61⋊C3)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC5×C17×C61C2×C3
LCSG ▷ C5×C17×C61
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 83
transitive degreeμt(G) = 5185

A minimal faithful permutation representation — 2 generators on 83 points:

(2,3)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(24,28,29,25,26,30)(27,36,37,31,32,46)(33,49,50,39,40,62)(34,48,44,42,38,35)(41,67,68,52,53,78)(43,65,66,54,51,59)(45,47,64,56,57,60)(55,80,81,70,69,76)(58,63,79,72,71,77)(61,83,82,74,75,73);(1,2,4,5,3)(6,7,9,11,13,15,17,19,21,22,20,18,16,14,12,10,8)(23,24,27,33,41,53,69,75,77,78,72,56,44,29,28,34,43,55,71,40,51,60,62,76,82,66,37,48,36,47,61,74,57,32,38,46,59,73,81,50,64,65,49,63,70,54,42,26,30,35,45,58,68,79,83,80,67,52,39,31,25)

Polycyclic presentation:

2,3,5,17,61|c^a=c^4,d^a=d^16,e^a=e^60,e^b=e^47

Conjugacy classes & character table

982 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31110, 33);