Finite Group DatabasebetaFinite Group Databasebeta| order | 55080 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D27540 |
| Fitting | C4×C5×C17×C81 ⋊ C2 |
| Hall | C5×C17×C81 ⋊ D4 |
| Hall | C17×C81 ⋊ D20 |
| Hall | C5×C81 ⋊ D68 |
| Hall | C5×C17 ⋊ D324 |
| Sylow | C81 ⋊ D340 |
| Sylow | C17 ⋊ D1620 |
| Sylow | C5 ⋊ D5508 |
160 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 160 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) = 27540 |
A minimal faithful permutation representation — 2 generators on 107 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);(2,4)(6,85)(7,84)(8,83)(9,82)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(87,90)(88,89)(92,107)(93,106)(94,105)(95,104)(96,103)(97,102)(98,101)(99,100)Polycyclic presentation:
2^3,3,5,17,3^3|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^16,g^a=g^2h^2i^2,h^a=h^2i^2,i^a=i^213773 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, 17, 3, 3, 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.6^16);;
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);