Finite Group DatabasebetaFinite Group Databasebeta| order | 3528 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C3×#3861 ⋊ D4 |
| Sylow | C72 ⋊ C3×D12 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×#64423 |
| commutator, Fitting, socle, cosocle | C2×C3×C72 | C22×C3 |
| hypocenter | C3×C72 | C3×D4 |
| LCS | G ▷ C2×C3×C72 ▷ C3×C72 | |
| UCS | C1 ◁ C2 | |
206 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 206 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:
2^2,3,2,3,7^2|a^2=d,b^a=bd,e^b=e^2,f^a=f^6,f^c=f^2,g^a=f^4g,g^c=g^287 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#11924207, #72222111, #72222115, … |
| maximal normal subgroup of | D3×#11924207, #72222111, #72222115, … |
| maximal quotient of | #72222162, C2×#72222053 |
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");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 3, 7, 7]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.6^4*F.7);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);