Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C9 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting, Sylow | #10624550 ⋊ C22 |
| derived | C92 ⋊ C22×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | #92264⋊C2 |
| commutator, hypocenter | C92 | C22×C9 |
| Frattini, socle | C33 | C3×D32 |
| Fitting | #10624550 | C22 |
| prime radical | C3×D92 | C3 |
| cosocle | C3×C92 | C22×C3 |
| LCS | G ▷ C92 | |
| UCS | C1 ◁ C3 | |
232 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 44 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:
2^2,3^6|c^3=d,e^3=g,f^3=h,c^a=cg^2,e^a=e^2g^2,e^c=eg^2,f^b=f^2h^2,f^c=fh,g^a=g^2,h^b=h^2108 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/9)×D9⋊C9, #72243339, #72243342, … |
| maximal normal subgroup of | AGL(1,ℤ/9)×D9⋊C9, #72243339, #72243342, … |
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> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.7^2);;
gap> SetConjugate(coll, 6, 2, F.6^2*F.8^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);