Finite Group DatabasebetaFinite Group Databasebeta| order | 57120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D28560 |
224 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 224 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 28560 |
A minimal faithful permutation representation — 2 generators on 48 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,19)(21,24)(22,23)(26,31)(27,30)(28,29)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)Polycyclic presentation:
2^5,3,5,7,17|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^4,h^a=h^6,i^a=i^1614283 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, 2, 2, 3, 5, 7, 17]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
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);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 9, 1, F.9^16);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);