Finite Group Databasebeta

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Sp(2,𝔽29) #72209938

order24360
isoclinismisoclinic class of Sp(2,𝔽29)
automorphismAut(G) = PGL(2,𝔽29) (order 24360) · Inn(G) = PSL(2,𝔽29) · Out(G) = C2
homologyAbelianization (H1) = C1 · Schur multiplier (H2) = C1

Aliases

symplecticSp(2,𝔽29)
special linearSL(2,𝔽29)
special unitarySU(2,𝔽29)

Subgroups

subgroupG/H
center, Frattini, Fitting, socle, cosocle, hypercenterC2PSL(2,𝔽29)
commutator, hypocenterGC1
LCSG
UCSC1 ◁ C2

29 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)(16,33,26,40)(19,22,21,20)(31,77,61,78)(32,74,46,76)(34,65,62,37)(35,84,63,85)(36,72,64,48)(38,91,42,82)(39,70,66,45)(41,95,67,96)(43,99,68,100)(44,47,69,71)(49,103,55,107)(50,60,56,54)(51,59,57,53)(52,108,58,89)(73,98,75,92)(79,114,113,86)(80,94,104,81)(83,109,111,90)(87,118,106,117)(88,116,115,97)(93,120,101,119)(102,105,112,110);(2,7,13,31,56,90,102,54,87,72,99,91,114,70,107,108,64,115,74,85,66,101,51,83,105,59,41,16,8)(3,11,23,61,50,109,112,60,106,48,100,82,86,45,103,89,36,88,76,84,39,93,57,111,110,53,67,26,12)(9,19,49,35,14,34,71,28,32,80,96,119,79,73,29,40,94,97,117,77,98,42,25,65,44,17,43,52,20)(10,21,55,63,24,62,47,18,46,104,95,120,113,75,30,33,81,116,118,78,92,38,15,37,69,27,68,58,22);(1,4,6,5)(2,9,3,10)(7,14,36,47,105,80,60,22,59,73,111,62,66,25,11,24,64,71,110,104,54,20,53,75,83,34,39,15)(8,17,45,37,90,98,57,21,56,94,112,69,72,28,12,27,70,65,109,92,51,19,50,81,102,44,48,18)(13,32,61,113,85,117,103,58,101,43,88,95,91,40,23,46,31,79,84,118,107,52,93,68,115,96,82,33)(16,42,67,116,99,119,89,55,87,35,86,78,74,29,26,38,41,97,100,120,108,49,106,63,114,77,76,30)

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

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)(16,33,26,40)(19,22,21,20)(31,77,61,78)(32,74,46,76)(34,65,62,37)(35,84,63,85)(36,72,64,48)(38,91,42,82)(39,70,66,45)(41,95,67,96)(43,99,68,100)(44,47,69,71)(49,103,55,107)(50,60,56,54)(51,59,57,53)(52,108,58,89)(73,98,75,92)(79,114,113,86)(80,94,104,81)(83,109,111,90)(87,118,106,117)(88,116,115,97)(93,120,101,119)(102,105,112,110), (2,7,13,31,56,90,102,54,87,72,99,91,114,70,107,108,64,115,74,85,66,101,51,83,105,59,41,16,8)(3,11,23,61,50,109,112,60,106,48,100,82,86,45,103,89,36,88,76,84,39,93,57,111,110,53,67,26,12)(9,19,49,35,14,34,71,28,32,80,96,119,79,73,29,40,94,97,117,77,98,42,25,65,44,17,43,52,20)(10,21,55,63,24,62,47,18,46,104,95,120,113,75,30,33,81,116,118,78,92,38,15,37,69,27,68,58,22), (1,4,6,5)(2,9,3,10)(7,14,36,47,105,80,60,22,59,73,111,62,66,25,11,24,64,71,110,104,54,20,53,75,83,34,39,15)(8,17,45,37,90,98,57,21,56,94,112,69,72,28,12,27,70,65,109,92,51,19,50,81,102,44,48,18)(13,32,61,113,85,117,103,58,101,43,88,95,91,40,23,46,31,79,84,118,107,52,93,68,115,96,82,33)(16,42,67,116,99,119,89,55,87,35,86,78,74,29,26,38,41,97,100,120,108,49,106,63,114,77,76,30));