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 | H(3,𝔽3)×3−1+2 ⋊ C22 |
| derived | C9×H(3,𝔽3) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C9×H(3,𝔽3) | C22×C3 |
| Frattini, socle | C32 | C3×D3×C3⋊D3 |
| Fitting | H(3,𝔽3)×3−1+2 | C22 |
| LCS | G ▷ C9×H(3,𝔽3) | |
| UCS | C1 | |
673 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 673 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
Polycyclic presentation:
2^2,3^6|f^3=g,d^a=d^2,e^a=e^2,e^b=e^2,e^d=eh,f^a=f^2g^2,f^c=fg,g^a=g^2,h^b=h^273 conjugacy classes — view the full character table.
| maximal subgroup of | #72253809, #72253822, #72253826, … |
| maximal normal subgroup of | C3×#72253807 |
| maximal quotient of | #72253808, #72253809, #72253826, … |
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, 6, F.7);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.7^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);