Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C24×C32 |
No aliases for this group.
| Fitting, derived | C24×C33 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C24×C33 | C2×C3 |
| Frattini | C3 | C3⋊D3×C22⋊A4 |
| socle | C24×C32 | C3×D3 |
| LCS | G ▷ C24×C33 | |
| UCS | C1 | |
3829 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 3829 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2,3,2^4,3^3|c^b=cd,d^b=c,e^b=f,f^b=ef,g^a=g^2,g^b=gi,h^a=h^2,i^a=i^296 conjugacy classes — view the full character table.
| maximal subgroup of | #72220505, #72220612 |
| maximal normal subgroup of | #72220505, #72220612 |
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, 2, 2, 2, 2, 3, 3, 3]);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 2, F.3);;
gap> SetConjugate(coll, 5, 2, F.6);;
gap> SetConjugate(coll, 6, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);