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 | C3×H(5,𝔽3) ⋊ C22 |
| derived | C32×H(3,𝔽3) ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C32×H(3,𝔽3) | C22×C3 |
| Frattini | C3 | C3×C3⋊#1113 |
| Fitting | C3×H(5,𝔽3) | C22 |
| socle | C32 | C3×#1113 |
| LCS | G ▷ C32×H(3,𝔽3) | |
| UCS | C1 | |
1576 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 1576 conjugacy classes of subgroups — too many to draw. Only its 72 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,d^c=dh,e^b=e^2,f^a=f^2,f^b=f^2h,f^e=fh,g^a=g^2,g^b=g^2,g^e=gh^2,h^a=h^2102 conjugacy classes — view the full character table.
| maximal subgroup of | #72221959, #72221963, #72243995, … |
| maximal normal subgroup of | D3×#66432⋊C3, #72277838 |
| maximal quotient of | #72277687 |
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, 4, 3, F.4*F.8);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 2, F.6^2*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 7, 5, F.7*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);