Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting, Sylow | #10624930 ⋊ C22 |
| derived | C3×AΣL(1,𝔽27) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×AΣL(1,𝔽27) | C22×C3 |
| Frattini | C32 | C3×D3×C3⋊D3 |
| Fitting | #10624930 | C22 |
| socle | C3 | C3×((C3⋊D3)⋊C3)⋊D3 |
| LCS | G ▷ C3×AΣL(1,𝔽27) | |
| UCS | C1 | |
645 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 645 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^2,3^6|d^a=d^2,e^a=e^2,e^c=eh,f^b=f^2,f^d=fg,f^e=fg^2,g^a=g^2,g^b=g^2h,g^f=gh,h^a=h^269 conjugacy classes — view the full character table.
| maximal subgroup of | #72221959, #72243790, #72243811, … |
| maximal normal subgroup of | #72243811, C3×#72221953, #72244021, … |
| maximal quotient of | #72221959, C3×#72221953, #72243931, … |
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");;
gap> coll := SingleCollector(F, [2, 2, 3, 3, 3, 3, 3, 3]);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.7^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.7^2*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);