Finite Group DatabasebetaFinite Group Databasebeta| order | 38880 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D19440 |
| Fitting | C5×C16×C243 ⋊ C2 |
| Hall | C5×C243 ⋊ D16 |
| Sylow | C243 ⋊ D80 |
| Sylow | C5 ⋊ D3888 |
168 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 168 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 264 |
| transitive degree | μt(G) = 19440 |
A minimal faithful permutation representation — 2 generators on 264 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259)(260,261,262,263,264);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,259)(19,258)(20,257)(21,256)(22,255)(23,254)(24,253)(25,252)(26,251)(27,250)(28,249)(29,248)(30,247)(31,246)(32,245)(33,244)(34,243)(35,242)(36,241)(37,240)(38,239)(39,238)(40,237)(41,236)(42,235)(43,234)(44,233)(45,232)(46,231)(47,230)(48,229)(49,228)(50,227)(51,226)(52,225)(53,224)(54,223)(55,222)(56,221)(57,220)(58,219)(59,218)(60,217)(61,216)(62,215)(63,214)(64,213)(65,212)(66,211)(67,210)(68,209)(69,208)(70,207)(71,206)(72,205)(73,204)(74,203)(75,202)(76,201)(77,200)(78,199)(79,198)(80,197)(81,196)(82,195)(83,194)(84,193)(85,192)(86,191)(87,190)(88,189)(89,188)(90,187)(91,186)(92,185)(93,184)(94,183)(95,182)(96,181)(97,180)(98,179)(99,178)(100,177)(101,176)(102,175)(103,174)(104,173)(105,172)(106,171)(107,170)(108,169)(109,168)(110,167)(111,166)(112,165)(113,164)(114,163)(115,162)(116,161)(117,160)(118,159)(119,158)(120,157)(121,156)(122,155)(123,154)(124,153)(125,152)(126,151)(127,150)(128,149)(129,148)(130,147)(131,146)(132,145)(133,144)(134,143)(135,142)(136,141)(137,140)(138,139)(261,264)(262,263)Polycyclic presentation:
2^5,3,5,3^4|b^2=cd,c^2=de,d^2=e,f^3=h,h^3=i,i^3=j,j^3=k,b^a=bc,c^a=cd,d^a=de,f^a=f^2h^2i^2j^2k^2,g^a=g^4,h^a=h^2i^2j^2k^2,i^a=i^2j^2k^2,j^a=j^2k^2,k^a=k^29723 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:
gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 5, 3, 3, 3, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetPower(coll, 9, F.10);;
gap> SetPower(coll, 10, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.8^2*F.9^2*F.10^2*F.11^2);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2*F.10^2*F.11^2);;
gap> SetConjugate(coll, 9, 1, F.9^2*F.10^2*F.11^2);;
gap> SetConjugate(coll, 10, 1, F.10^2*F.11^2);;
gap> SetConjugate(coll, 11, 1, F.11^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);