Finite Group DatabasebetaFinite Group Databasebeta| order | 3888 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
392 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,3,4)(5,6,8,7)(9,15,16,11,10,17,14,12,13)(18,19)(20,26)(22,23)(24,25);(6,7,8)(10,12,15)(13,17,16)(18,24,25,21,22,26,19,20,23)Polycyclic presentation:
2,3^2,2,3^3,2^2|a^2=d,c^3=e,f^3=g,c^a=cg^2,c^b=ce,f^a=f^2g^2,f^b=fg,g^a=g^2,h^a=i,h^f=i,i^a=h,i^f=hi150 conjugacy classes — view the full character table.
| maximal subgroup of | #72245336 |
| maximal quotient of | #72245336 |
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");;
gap> coll := SingleCollector(F, [2, 3, 3, 2, 3, 3, 3, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.9);;
gap> SetConjugate(coll, 8, 6, F.9);;
gap> SetConjugate(coll, 9, 1, F.8);;
gap> SetConjugate(coll, 9, 6, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);