Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C2×C3×#2010 ⋊ C2 |
| Sylow | C3 ⋊ #417674 |
5826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 23 points:
(2,3)(9,16)(10,19)(11,13)(14,22)(15,17)(18,23);(1,2,3)(5,7)(9,15)(16,17);(8,10,11,20,12,19,13,21)(9,14,16,18,15,23,17,22);(4,5)(6,7)(8,9)(10,18)(11,17)(12,15)(13,16)(14,20)(19,22)(21,23)Polycyclic presentation:
2^9,3|b^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,d^a=dfhi,d^b=defhi,e^a=ei,e^d=ei,f^b=fi,g^a=gi,h^a=hi,h^d=hi,j^d=j^2114 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#417674, #72254962, #72269869, … |
| maximal normal subgroup of | D3×#417674, #72254962, #72269869, … |
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, 2, F.5);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.6*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, 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, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 10, 4, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);