Finite Group DatabasebetaFinite Group Databasebeta| order | 3888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C4×3−1+4 ⋊ C22 |
| Sylow | 3−1+4 ⋊ C2×D4 |
971 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 971 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 108 |
Polycyclic presentation:
2^3,3,2,3^4|a^2=e,g^3=i,b^a=be,d^b=di^2,f^b=f^2,f^c=f^2,g^b=g^2i^2,g^d=gi^2,h^c=h^2,h^f=hi^2,i^b=i^2152 conjugacy classes — view the full character table.
| maximal subgroup of | D4×(#4207⋊C2)⋊C3, #72276036 |
| maximal normal subgroup of | D4×(#4207⋊C2)⋊C3 |
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, 2, 3, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.9^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.7^2*F.9^2);;
gap> SetConjugate(coll, 7, 4, F.7*F.9^2);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 8, 6, F.8*F.9^2);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);