Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| Fitting | C2×C3×#3025 ⋊ C2 |
| Sylow | C3 ⋊ #498888 |
4668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 23 points:
(5,7,9,14)(6,11,12,19)(8,16,17,15)(10,18,13,20)(21,22,23);(2,4)(6,10)(7,14)(11,18)(12,13)(15,16)(19,20)(22,23);(1,2)(3,4)(10,13)(18,20);(5,6)(7,11)(8,10,17,13)(9,12)(14,19)(15,20,16,18)Polycyclic presentation:
2^9,3|a^2=f,d^2=i,g^2=i,b^a=bf,c^a=cfgi,c^b=ce,d^c=dhi,e^a=ei,f^c=fh,g^a=gh,g^b=gh,g^c=gh,g^d=gi,g^f=gi,h^a=hi,j^c=j^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #72248020, #72254161, D3×#498888 |
| maximal normal subgroup of | #72248020, #72254161, D3×#498888 |
| maximal quotient of | #72271616 |
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, 4, F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.7*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 10, 3, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);