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C3⋊(C17⋊(D61⋊C5)) #44527453

order31110
automorphism?
homologyAbelianization (H1) = C2×C5 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productC3⋊(C17⋊(D61⋊C5))
unambiguous semidirect productC17⋊(C3⋊(D61⋊C5))
unambiguous semidirect product(C61⋊C5)⋊D51
unambiguous semidirect productD3111⋊C5

Splittings

Fitting, socle, derived, HallC3×C17×C61 ⋊ C2×C5
HallC3×C17×C61⋊C5 ⋊ C2
HallD3111 ⋊ C5
HallC17×C61⋊C5 ⋊ D3
HallC17×C61 ⋊ C5×D3
HallC3×C61⋊C5 ⋊ D17
HallC61⋊C5 ⋊ D51
HallC3×C61 ⋊ C5×D17
SylowC61 ⋊ C5×D51
HallC3×C17 ⋊ D61⋊C5
SylowC17 ⋊ C3⋊(D61⋊C5)
SylowC3 ⋊ C17⋊(D61⋊C5)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C17×C61C2×C5
LCSG ▷ C3×C17×C61
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 81
transitive degreeμt(G) = 3111

A minimal faithful permutation representation — 2 generators on 81 points:

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

Polycyclic presentation:

2,3,5,17,61|b^a=b^2,d^a=d^16,e^a=e^60,e^c=e^9

Conjugacy classes & character table

441 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31110, 26);