Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting, Sylow | #392611 ⋊ C3 |
1270 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 1270 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,3,4,2)(6,11)(8,16)(9,18,22,17,10,21)(12,13)(14,28,30,27,15,35)(19,20,34)(23,26,36,25,24,29)(31,32,33);(1,2,4,3)(5,29)(6,22)(7,30)(8,35)(9,26)(10,14)(11,34)(12,21)(13,33)(15,17)(16,36)(18,25)(19,28)(20,23)(24,31)(27,32)Polycyclic presentation:
2^2,3,2^7|a^2=d,b^2=dj,e^2=j,f^2=j,g^2=j,b^a=be,e^a=ej,e^b=ej,f^b=fi,f^c=fg,g^b=gh,g^c=f,g^f=gj,h^c=i,h^f=hj,i^c=hij,i^g=ij68 conjugacy classes — view the full character table.
| maximal subgroup of | #72341828, #72341898, #72341905, … |
| maximal normal subgroup of | #72341828, #72341898, #72341905, … |
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, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4*F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, F.6);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);