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D51×D31⋊C5 #44539293

order31620
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2

Aliases

direct productD51×D31⋊C5
unambiguous semidirect productD31⋊(C5×D51)
unambiguous semidirect product(C3×D31⋊C5)⋊D17
unambiguous semidirect product(C17×D31⋊C5)⋊D3
unambiguous semidirect product(D31×D51)⋊C5

Splittings

Fitting, socle, derived, HallC3×C17×C31 ⋊ C22×C5
HallC3×C17×ΓL(1,𝔽32) ⋊ C22
HallD31×D51 ⋊ C5
HallC17×ΓL(1,𝔽32) ⋊ C2×D3
HallC17×C31 ⋊ C2×C5×D3
HallC3×ΓL(1,𝔽32) ⋊ C2×D17
HallΓL(1,𝔽32) ⋊ C2×D51
HallC3×C31 ⋊ C2×C5×D17
HallC3×C17 ⋊ C2×D31⋊C5
SylowC31 ⋊ C2×C5×D51
SylowC17 ⋊ D3×D31⋊C5
SylowC3 ⋊ D17×D31⋊C5

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC3×C17×C31C22×C5
LCSG ▷ C3×C17×C31
UCSC1

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 51
transitive degreeμt(G) = 1581

A minimal faithful permutation representation — 2 generators on 51 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31620, 50);