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Sp(2,𝔽13) #72209592

order2184
isoclinismisoclinic class of Sp(2,𝔽13)
automorphismAut(G) = PGL(2,𝔽13) (order 2184) · Inn(G) = PSL(2,𝔽13) · Out(G) = C2
homologyAbelianization (H1) = C1 · Schur multiplier (H2) = C1

Aliases

symplecticSp(2,𝔽13)
special linearSL(2,𝔽13)
special unitarySU(2,𝔽13)

Subgroups

subgroupG/H
center, Frattini, Fitting, socle, cosocle, hypercenterC2PSL(2,𝔽13)
commutator, hypocenterGC1
LCSG
UCSC1 ◁ C2

21 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Permutation generators:

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

Conjugacy classes & character table

17 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC7×Sp(2,𝔽13), #72234266, C7×#72234266
maximal subgroup ofC3×Sp(2,𝔽13), #72210876, C7×Sp(2,𝔽13), …
maximal normal subgroup ofC3×Sp(2,𝔽13), #72210876, C7×Sp(2,𝔽13), …
maximal quotient ofC3×Sp(2,𝔽13), C7×Sp(2,𝔽13)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:

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