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×#1891 ⋊ C2 |
| Sylow | C3 ⋊ #147817 |
2412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(2,4)(5,7)(6,12)(8,16)(9,18)(10,17)(11,19)(13,15)(14,20)(22,23);(1,2)(3,4)(6,11)(12,14)(13,20)(15,19)(21,22,23);(5,6,8,12,10,13,18,15)(7,11,9,14,17,20,16,19)Polycyclic presentation:
2^9,3|a^2=e,e^2=i,f^2=i,g^2=i,b^a=befi,c^a=cefg,c^b=cd,d^a=di,e^c=ei,f^c=fh,g^a=gi,g^b=ghi,g^c=ghi,g^f=gi,h^a=hi,j^c=j^2120 conjugacy classes — view the full character table.
| maximal subgroup of | #72248020, #72254126, D3×#147817 |
| maximal normal subgroup of | #72248020, #72254126, D3×#147817 |
| maximal quotient of | #72271617 |
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.5);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.6*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*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);