Finite Group DatabasebetaFinite Group Databasebeta| order | 3888 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | #6900 ⋊ C22⋊C4 |
593 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 593 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^2,3,2^2,3^4|a^2=d,f^3=h,g^3=i,b^a=bde,f^a=f^2h^2,f^b=f^2h^2,f^c=fh^2,g^a=f^2g,g^b=g^2i^2,g^c=gi^2,h^a=h^2,h^b=h^2,i^a=h^2i,i^b=i^2126 conjugacy classes — view the full character table.
| maximal subgroup of | #72220027, #72220044, #72220053, … |
| maximal normal subgroup of | #72220027, #72220044, #72220053, … |
| maximal quotient of | #72219983, #72220088 |
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, 2, 3, 2, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.8^2);;
gap> SetConjugate(coll, 6, 2, F.6^2*F.8^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.8^2);;
gap> SetConjugate(coll, 7, 1, F.6^2*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^2*F.9^2);;
gap> SetConjugate(coll, 7, 3, F.7*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);