Finite Group DatabasebetaFinite Group Databasebeta| order | 42120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| dicyclic | Dic10530 |
| Hall | C5×C13×C81 ⋊ Q8 |
| Hall | C13×C81 ⋊ Dic10 |
| Hall | C5×C81 ⋊ Dic26 |
| Sylow | C81 ⋊ Dic130 |
| Hall | C5×C13 ⋊ Dic162 |
| Sylow | C13 ⋊ Dic810 |
| Sylow | C5 ⋊ Dic2106 |
120 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 120 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) = 107 |
| transitive degree | μt(G) = 42120 |
A minimal faithful permutation representation — 2 generators on 107 points:
(1,2,3,4)(5,8,7,6)(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);(1,5,3,7)(2,6,4,8)(10,89)(11,88)(12,87)(13,86)(14,85)(15,84)(16,83)(17,82)(18,81)(19,80)(20,79)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(91,94)(92,93)(96,107)(97,106)(98,105)(99,104)(100,103)(101,102)Polycyclic presentation:
2^3,3,5,13,3^3|a^2=c,b^2=c,d^3=g,g^3=h,h^3=i,b^a=bc,d^a=d^2g^2h^2i^2,e^a=e^4,f^a=f^12,g^a=g^2h^2i^2,h^a=h^2i^2,i^a=i^210533 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 13, 3, 3, 3]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^12);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);