Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C22×C32 |
No aliases for this group.
| Fitting | C24×C32 ⋊ C3×D3 |
| derived | C22×C32×A4 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×C32×A4 | C2×C3 |
| Frattini | C3 | A4×C3⋊S4 |
| Fitting | C24×C32 | C3×D3 |
| socle | C24×C3 | C3×C3⋊D3 |
| LCS | G ▷ C22×C32×A4 | |
| UCS | C1 | |
669 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 669 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2,3,2^2,3^3,2^2|c^b=cd,d^b=c,e^a=e^2,e^b=eg,f^a=f^2,g^a=g^2,h^f=i,i^a=hi,i^f=hi52 conjugacy classes — view the full character table.
| maximal subgroup of | S4×(C3⋊D3)⋊A4, #72222956, #72222975 |
| maximal normal subgroup of | S4×(C3⋊D3)⋊A4, #72222956, #72222975 |
| maximal quotient of | #72220531 |
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, 3, 3, 3, 2, 2]);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 2, F.3);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 6, F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);