Finite Group DatabasebetaFinite Group Databasebeta| order | 15120 |
| isoclinism | isoclinic class of #72209688 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, Fitting, socle, cosocle, hypercenter | C2×C3 | A7 |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 ◁ C2×C3 | |
92 conjugacy classes of subgroups.
Permutation generators:
(1,2,3)(4,10,8,15,6,11)(5,13,9,17,7,14)(12,16,18)(19,31,27,36,21,32)(20,34,28,38,22,35)(23,33,29,41,25,37)(24,39,30,42,26,40)(43,44,45)(46,47,50,48)(49,53,55,54)(51,56,52,57)(58,66,60,67)(59,69,61,70)(62,72,64,73)(63,75,65,76)(68,77,71,74)(78,94,80,95)(79,97,81,98)(82,100,84,101)(83,103,85,104)(86,106,88,107)(87,109,89,110)(90,112,92,113)(91,115,93,116)(96,120,99,121)(102,130,105,131)(108,140,111,141)(114,150,117,151)(118,143,119,145)(122,158,124,159)(123,161,125,162)(126,152,128,153)(127,163,129,160)(132,166,134,167)(133,138,135,136)(137,171,139,172)(142,176,144,177)(146,180,148,181)(147,157,149,155)(154,182,156,183)(164,186,165,187)(168,192,169,193)(170,194,173,195)(174,200,175,201)(178,202,179,203)(184,210,185,211)(188,216,190,217)(189,219,191,220)(196,226,198,227)(197,229,199,230)(204,236,205,237)(206,238,208,239)(207,240,209,241)(212,242,214,243)(213,244,215,245)(218,231,221,228)(222,252,224,253)(223,254,225,255)(232,264,234,265)(233,259,235,257)(246,267,248,269)(247,270,249,271)(250,272,251,273)(256,274,258,275)(260,266,262,268)(261,276,263,277)(278,285,279,284)(280,283,281,282);(1,4,12,5)(2,6,16,7)(3,8,18,9)(10,19,33,20)(11,21,37,22)(13,23,39,24)(14,25,40,26)(15,27,41,28)(17,29,42,30)(34,43)(35,44)(38,45)(46,49,52,48,50,55,51,47)(53,58,68,61,54,60,71,59)(56,62,74,65,57,64,77,63)(66,78,96,81,67,80,99,79)(69,82,102,85,70,84,105,83)(72,86,108,89,73,88,111,87)(75,90,114,93,76,92,117,91)(94,115,152,119,95,116,153,118)(97,122,160,125,98,124,163,123)(100,126,164,129,101,128,165,127)(103,132,168,135,104,134,169,133)(106,136,170,139,107,138,173,137)(109,142,178,145,110,144,179,143)(112,146,177,149,113,148,176,147)(120,154,130,157,121,156,131,155)(140,174,150,158,141,175,151,159)(161,184,204,180,162,185,205,181)(166,188,218,191,167,190,221,189)(171,196,228,199,172,198,231,197)(182,206,211,209,183,208,210,207)(186,212,192,215,187,214,193,213)(194,222,202,225,195,224,203,223)(200,232,237,235,201,234,236,233)(216,246,255,249,217,248,254,247)(219,238,253,251,220,239,252,250)(226,256,243,259,227,258,242,257)(229,260,244,263,230,262,245,261)(240,266,278,269,241,268,279,267)(264,276,280,270,265,277,281,271)(272,274,284,283,273,275,285,282);(46,50)(47,48)(49,55)(51,52)(53,54)(56,57)(58,60)(59,61)(62,64)(63,65)(66,67)(68,71)(69,70)(72,73)(74,77)(75,76)(78,80)(79,81)(82,84)(83,85)(86,88)(87,89)(90,92)(91,93)(94,95)(96,99)(97,98)(100,101)(102,105)(103,104)(106,107)(108,111)(109,110)(112,113)(114,117)(115,116)(118,119)(120,121)(122,124)(123,125)(126,128)(127,129)(130,131)(132,134)(133,135)(136,138)(137,139)(140,141)(142,144)(143,145)(146,148)(147,149)(150,151)(152,153)(154,156)(155,157)(158,159)(160,163)(161,162)(164,165)(166,167)(168,169)(170,173)(171,172)(174,175)(176,177)(178,179)(180,181)(182,183)(184,185)(186,187)(188,190)(189,191)(192,193)(194,195)(196,198)(197,199)(200,201)(202,203)(204,205)(206,208)(207,209)(210,211)(212,214)(213,215)(216,217)(218,221)(219,220)(222,224)(223,225)(226,227)(228,231)(229,230)(232,234)(233,235)(236,237)(238,239)(240,241)(242,243)(244,245)(246,248)(247,249)(250,251)(252,253)(254,255)(256,258)(257,259)(260,262)(261,263)(264,265)(266,268)(267,269)(270,271)(272,273)(274,275)(276,277)(278,279)(280,281)(282,283)(284,285)40 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its permutation generators:
gap> G := Group((1,2,3)(4,10,8,15,6,11)(5,13,9,17,7,14)(12,16,18)(19,31,27,36,21,32)(20,34,28,38,22,35)(23,33,29,41,25,37)(24,39,30,42,26,40)(43,44,45)(46,47,50,48)(49,53,55,54)(51,56,52,57)(58,66,60,67)(59,69,61,70)(62,72,64,73)(63,75,65,76)(68,77,71,74)(78,94,80,95)(79,97,81,98)(82,100,84,101)(83,103,85,104)(86,106,88,107)(87,109,89,110)(90,112,92,113)(91,115,93,116)(96,120,99,121)(102,130,105,131)(108,140,111,141)(114,150,117,151)(118,143,119,145)(122,158,124,159)(123,161,125,162)(126,152,128,153)(127,163,129,160)(132,166,134,167)(133,138,135,136)(137,171,139,172)(142,176,144,177)(146,180,148,181)(147,157,149,155)(154,182,156,183)(164,186,165,187)(168,192,169,193)(170,194,173,195)(174,200,175,201)(178,202,179,203)(184,210,185,211)(188,216,190,217)(189,219,191,220)(196,226,198,227)(197,229,199,230)(204,236,205,237)(206,238,208,239)(207,240,209,241)(212,242,214,243)(213,244,215,245)(218,231,221,228)(222,252,224,253)(223,254,225,255)(232,264,234,265)(233,259,235,257)(246,267,248,269)(247,270,249,271)(250,272,251,273)(256,274,258,275)(260,266,262,268)(261,276,263,277)(278,285,279,284)(280,283,281,282), (1,4,12,5)(2,6,16,7)(3,8,18,9)(10,19,33,20)(11,21,37,22)(13,23,39,24)(14,25,40,26)(15,27,41,28)(17,29,42,30)(34,43)(35,44)(38,45)(46,49,52,48,50,55,51,47)(53,58,68,61,54,60,71,59)(56,62,74,65,57,64,77,63)(66,78,96,81,67,80,99,79)(69,82,102,85,70,84,105,83)(72,86,108,89,73,88,111,87)(75,90,114,93,76,92,117,91)(94,115,152,119,95,116,153,118)(97,122,160,125,98,124,163,123)(100,126,164,129,101,128,165,127)(103,132,168,135,104,134,169,133)(106,136,170,139,107,138,173,137)(109,142,178,145,110,144,179,143)(112,146,177,149,113,148,176,147)(120,154,130,157,121,156,131,155)(140,174,150,158,141,175,151,159)(161,184,204,180,162,185,205,181)(166,188,218,191,167,190,221,189)(171,196,228,199,172,198,231,197)(182,206,211,209,183,208,210,207)(186,212,192,215,187,214,193,213)(194,222,202,225,195,224,203,223)(200,232,237,235,201,234,236,233)(216,246,255,249,217,248,254,247)(219,238,253,251,220,239,252,250)(226,256,243,259,227,258,242,257)(229,260,244,263,230,262,245,261)(240,266,278,269,241,268,279,267)(264,276,280,270,265,277,281,271)(272,274,284,283,273,275,285,282), (46,50)(47,48)(49,55)(51,52)(53,54)(56,57)(58,60)(59,61)(62,64)(63,65)(66,67)(68,71)(69,70)(72,73)(74,77)(75,76)(78,80)(79,81)(82,84)(83,85)(86,88)(87,89)(90,92)(91,93)(94,95)(96,99)(97,98)(100,101)(102,105)(103,104)(106,107)(108,111)(109,110)(112,113)(114,117)(115,116)(118,119)(120,121)(122,124)(123,125)(126,128)(127,129)(130,131)(132,134)(133,135)(136,138)(137,139)(140,141)(142,144)(143,145)(146,148)(147,149)(150,151)(152,153)(154,156)(155,157)(158,159)(160,163)(161,162)(164,165)(166,167)(168,169)(170,173)(171,172)(174,175)(176,177)(178,179)(180,181)(182,183)(184,185)(186,187)(188,190)(189,191)(192,193)(194,195)(196,198)(197,199)(200,201)(202,203)(204,205)(206,208)(207,209)(210,211)(212,214)(213,215)(216,217)(218,221)(219,220)(222,224)(223,225)(226,227)(228,231)(229,230)(232,234)(233,235)(236,237)(238,239)(240,241)(242,243)(244,245)(246,248)(247,249)(250,251)(252,253)(254,255)(256,258)(257,259)(260,262)(261,263)(264,265)(266,268)(267,269)(270,271)(272,273)(274,275)(276,277)(278,279)(280,281)(282,283)(284,285));