Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C2×C3×#1781 ⋊ C2 |
| Sylow | C3 ⋊ #147567 |
4902 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,3)(5,7)(11,21)(13,22)(15,19)(18,20);(8,10)(9,13)(11,20,21,18)(12,17)(14,22)(15,23,19,16);(2,3)(4,5,6,7)(8,9)(10,16,17,23)(11,19)(12,14)(13,18,22,20)(15,21)Polycyclic presentation:
2^9,3|a^2=f,b^2=d,b^a=bg,c^a=cfh,c^b=ce,d^a=dh,e^a=ei,f^b=fi,g^a=gi,g^b=gh,g^c=gi,g^d=gi,h^b=hi,j^a=j^2120 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#147567, #72248495, #72248542 |
| maximal normal subgroup of | D3×#147567, #72248495, #72248542 |
| maximal quotient of | #72255199 |
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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 3]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);