Sp(2,𝔽₁₇) #72209610

order	4896
isoclinism	isoclinic class of Sp(2,𝔽₁₇)
automorphism	Aut(G) = PGL(2,𝔽₁₇) (order 4896) · Inn(G) = PSL(2,𝔽₁₇) · Out(G) = C₂
homology	Abelianization (H₁) = C₁ · Schur multiplier (H₂) = C₁

Aliases

symplectic	Sp(2,𝔽₁₇)
special linear	SL(2,𝔽₁₇)
special unitary	SU(2,𝔽₁₇)

Subgroups

	subgroup	G/H
center, Frattini, Fitting, socle, cosocle, hypercenter	C₂	PSL(2,𝔽₁₇)
commutator, hypocenter	G	C₁
LCS	G
UCS	C₁ ◁ C₂

26 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Permutation generators:

(1,2,6,3)(4,10,16,14)(5,9,18,13)(7,12,11,8)(15,30,44,42)(17,27,46,39)(19,47,31,48)(20,36,32,24)(21,35,33,23)(22,55,34,56)(25,38,37,26)(28,41,40,29)(43,74,100,98)(45,67,102,91)(49,107,75,108)(50,104,76,106)(51,113,77,114)(52,86,78,62)(53,120,79,121)(54,84,80,60)(57,110,81,130)(58,127,82,129)(59,134,83,135)(61,123,85,139)(63,143,87,144)(64,146,88,147)(65,90,89,66)(68,95,92,71)(69,94,93,70)(72,97,96,73)(99,164,101,153)(103,177,105,132)(109,213,165,214)(111,126,166,128)(112,205,167,210)(115,137,168,180)(116,220,169,221)(117,136,170,179)(118,225,171,226)(119,184,172,142)(122,232,173,233)(124,236,174,237)(125,181,175,138)(131,245,176,246)(133,240,178,243)(140,235,182,252)(141,238,183,253)(145,256,185,217)(148,248,186,258)(149,259,187,260)(150,261,188,262)(151,190,189,152)(154,194,191,157)(155,193,192,156)(158,198,195,161)(159,197,196,160)(162,200,199,163)(201,229,206,242)(202,251,207,219)(203,218,208,231)(204,227,209,241)(211,281,212,282)(215,249,265,257)(216,283,244,284)(222,285,266,286)(223,280,267,276)(224,247,255,270)(228,288,268,287)(230,250,239,254)(234,279,269,275)(263,278,271,274)(264,273,272,277);(2,7,19,49,109,215,248,140,70,68,122,217,247,131,57,22,8)(3,11,31,75,165,265,258,182,94,92,173,256,270,176,81,34,12)(9,25,63,145,156,234,136,59,23,20,51,115,222,154,148,64,26)(10,28,21,53,50,111,110,216,160,158,212,108,132,58,61,24,29)(13,37,87,185,193,269,179,83,35,32,77,168,266,191,186,88,38)(14,40,33,79,76,166,130,244,197,195,211,107,177,82,85,36,41)(27,65,149,137,60,116,206,105,48,55,126,227,118,52,117,150,66)(30,72,69,124,54,114,123,219,112,133,218,121,134,62,141,71,73)(39,89,187,180,84,169,201,103,47,56,128,241,171,78,170,188,90)(42,96,93,174,80,113,139,251,167,178,231,120,135,86,183,95,97)(67,151,138,209,106,208,228,224,144,146,249,223,207,127,229,119,152)(74,162,159,263,155,233,238,225,125,142,221,237,235,157,264,161,163)(91,189,181,204,104,203,268,255,143,147,257,267,202,129,242,172,190)(98,199,196,271,192,232,253,226,175,184,220,236,252,194,272,198,200)(153,250,210,280,214,282,277,286,260,261,279,278,284,245,287,240,230)(164,254,205,276,213,281,273,285,259,262,275,274,283,246,288,243,239);(1,4,15,43,99,102,46,18,6,16,44,100,101,45,17,5)(2,9,27,67,153,98,42,14,3,13,39,91,164,74,30,10)(7,20,52,119,230,175,80,33,11,32,78,172,239,125,54,21)(8,23,60,138,250,184,86,36,12,35,84,181,254,142,62,24)(19,25,55,127,240,200,167,76,31,37,56,129,243,163,112,50)(22,26,48,106,210,199,178,82,34,38,47,104,205,162,133,58)(28,68,154,150,152,260,192,93,40,92,191,188,190,259,155,69)(29,70,156,149,151,261,194,95,41,94,193,187,189,262,157,71)(49,51,116,223,287,226,135,130,75,77,169,267,288,225,134,110)(53,122,234,227,229,286,252,139,79,173,269,241,242,285,235,123)(57,59,118,228,280,220,113,107,81,83,171,268,276,221,114,108)(61,140,222,206,209,279,232,120,85,182,266,201,204,275,233,121)(63,65,146,245,198,97,197,165,87,89,147,246,161,73,160,109)(64,66,144,214,196,96,195,176,88,90,143,213,159,72,158,131)(103,202,273,263,218,111,217,148,105,207,277,271,231,166,256,186)(115,117,224,284,253,174,211,265,168,170,255,283,238,124,212,215)(126,208,278,272,251,177,258,185,128,203,274,264,219,132,248,145)(136,137,249,282,236,183,244,270,179,180,257,281,237,141,216,247);(1,6)(2,3)(4,16)(5,18)(7,11)(8,12)(9,13)(10,14)(15,44)(17,46)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(43,100)(45,102)(47,48)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,56)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,97)(74,98)(99,101)(103,105)(104,106)(107,108)(109,165)(110,130)(111,166)(112,167)(113,114)(115,168)(116,169)(117,170)(118,171)(119,172)(120,121)(122,173)(123,139)(124,174)(125,175)(126,128)(127,129)(131,176)(132,177)(133,178)(134,135)(136,179)(137,180)(138,181)(140,182)(141,183)(142,184)(143,144)(145,185)(146,147)(148,186)(149,187)(150,188)(151,189)(152,190)(153,164)(154,191)(155,192)(156,193)(157,194)(158,195)(159,196)(160,197)(161,198)(162,199)(163,200)(201,206)(202,207)(203,208)(204,209)(205,210)(211,212)(213,214)(215,265)(216,244)(217,256)(218,231)(219,251)(220,221)(222,266)(223,267)(224,255)(225,226)(227,241)(228,268)(229,242)(230,239)(232,233)(234,269)(235,252)(236,237)(238,253)(240,243)(245,246)(247,270)(248,258)(249,257)(250,254)(259,260)(261,262)(263,271)(264,272)(273,277)(274,278)(275,279)(276,280)(281,282)(283,284)(285,286)(287,288)

Conjugacy classes & character table

21 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)(4,10,16,14)(5,9,18,13)(7,12,11,8)(15,30,44,42)(17,27,46,39)(19,47,31,48)(20,36,32,24)(21,35,33,23)(22,55,34,56)(25,38,37,26)(28,41,40,29)(43,74,100,98)(45,67,102,91)(49,107,75,108)(50,104,76,106)(51,113,77,114)(52,86,78,62)(53,120,79,121)(54,84,80,60)(57,110,81,130)(58,127,82,129)(59,134,83,135)(61,123,85,139)(63,143,87,144)(64,146,88,147)(65,90,89,66)(68,95,92,71)(69,94,93,70)(72,97,96,73)(99,164,101,153)(103,177,105,132)(109,213,165,214)(111,126,166,128)(112,205,167,210)(115,137,168,180)(116,220,169,221)(117,136,170,179)(118,225,171,226)(119,184,172,142)(122,232,173,233)(124,236,174,237)(125,181,175,138)(131,245,176,246)(133,240,178,243)(140,235,182,252)(141,238,183,253)(145,256,185,217)(148,248,186,258)(149,259,187,260)(150,261,188,262)(151,190,189,152)(154,194,191,157)(155,193,192,156)(158,198,195,161)(159,197,196,160)(162,200,199,163)(201,229,206,242)(202,251,207,219)(203,218,208,231)(204,227,209,241)(211,281,212,282)(215,249,265,257)(216,283,244,284)(222,285,266,286)(223,280,267,276)(224,247,255,270)(228,288,268,287)(230,250,239,254)(234,279,269,275)(263,278,271,274)(264,273,272,277), (2,7,19,49,109,215,248,140,70,68,122,217,247,131,57,22,8)(3,11,31,75,165,265,258,182,94,92,173,256,270,176,81,34,12)(9,25,63,145,156,234,136,59,23,20,51,115,222,154,148,64,26)(10,28,21,53,50,111,110,216,160,158,212,108,132,58,61,24,29)(13,37,87,185,193,269,179,83,35,32,77,168,266,191,186,88,38)(14,40,33,79,76,166,130,244,197,195,211,107,177,82,85,36,41)(27,65,149,137,60,116,206,105,48,55,126,227,118,52,117,150,66)(30,72,69,124,54,114,123,219,112,133,218,121,134,62,141,71,73)(39,89,187,180,84,169,201,103,47,56,128,241,171,78,170,188,90)(42,96,93,174,80,113,139,251,167,178,231,120,135,86,183,95,97)(67,151,138,209,106,208,228,224,144,146,249,223,207,127,229,119,152)(74,162,159,263,155,233,238,225,125,142,221,237,235,157,264,161,163)(91,189,181,204,104,203,268,255,143,147,257,267,202,129,242,172,190)(98,199,196,271,192,232,253,226,175,184,220,236,252,194,272,198,200)(153,250,210,280,214,282,277,286,260,261,279,278,284,245,287,240,230)(164,254,205,276,213,281,273,285,259,262,275,274,283,246,288,243,239), (1,4,15,43,99,102,46,18,6,16,44,100,101,45,17,5)(2,9,27,67,153,98,42,14,3,13,39,91,164,74,30,10)(7,20,52,119,230,175,80,33,11,32,78,172,239,125,54,21)(8,23,60,138,250,184,86,36,12,35,84,181,254,142,62,24)(19,25,55,127,240,200,167,76,31,37,56,129,243,163,112,50)(22,26,48,106,210,199,178,82,34,38,47,104,205,162,133,58)(28,68,154,150,152,260,192,93,40,92,191,188,190,259,155,69)(29,70,156,149,151,261,194,95,41,94,193,187,189,262,157,71)(49,51,116,223,287,226,135,130,75,77,169,267,288,225,134,110)(53,122,234,227,229,286,252,139,79,173,269,241,242,285,235,123)(57,59,118,228,280,220,113,107,81,83,171,268,276,221,114,108)(61,140,222,206,209,279,232,120,85,182,266,201,204,275,233,121)(63,65,146,245,198,97,197,165,87,89,147,246,161,73,160,109)(64,66,144,214,196,96,195,176,88,90,143,213,159,72,158,131)(103,202,273,263,218,111,217,148,105,207,277,271,231,166,256,186)(115,117,224,284,253,174,211,265,168,170,255,283,238,124,212,215)(126,208,278,272,251,177,258,185,128,203,274,264,219,132,248,145)(136,137,249,282,236,183,244,270,179,180,257,281,237,141,216,247), (1,6)(2,3)(4,16)(5,18)(7,11)(8,12)(9,13)(10,14)(15,44)(17,46)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(43,100)(45,102)(47,48)(49,75)(50,76)(51,77)(52,78)(53,79)(54,80)(55,56)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96)(73,97)(74,98)(99,101)(103,105)(104,106)(107,108)(109,165)(110,130)(111,166)(112,167)(113,114)(115,168)(116,169)(117,170)(118,171)(119,172)(120,121)(122,173)(123,139)(124,174)(125,175)(126,128)(127,129)(131,176)(132,177)(133,178)(134,135)(136,179)(137,180)(138,181)(140,182)(141,183)(142,184)(143,144)(145,185)(146,147)(148,186)(149,187)(150,188)(151,189)(152,190)(153,164)(154,191)(155,192)(156,193)(157,194)(158,195)(159,196)(160,197)(161,198)(162,199)(163,200)(201,206)(202,207)(203,208)(204,209)(205,210)(211,212)(213,214)(215,265)(216,244)(217,256)(218,231)(219,251)(220,221)(222,266)(223,267)(224,255)(225,226)(227,241)(228,268)(229,242)(230,239)(232,233)(234,269)(235,252)(236,237)(238,253)(240,243)(245,246)(247,270)(248,258)(249,257)(250,254)(259,260)(261,262)(263,271)(264,272)(273,277)(274,278)(275,279)(276,280)(281,282)(283,284)(285,286)(287,288));

Sp(2,𝔽17) #72209610

Aliases

Subgroups

Presentation

Conjugacy classes & character table

GAP

Sp(2,𝔽₁₇) #72209610