Finite Group DatabasebetaFinite Group Databasebeta| order | 3402 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C33 |
No aliases for this group.
| Fitting | C3×C7×3−1+2 ⋊ C2×C3 |
| Hall | #13133543 ⋊ C2 |
| Hall | D7 ⋊ #6891 |
| Sylow | C7 ⋊ C2×#6891 |
504 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 504 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 189 |
Polycyclic presentation:
2,3^5,7|c^3=f,c^b=ce,d^c=df,g^a=g^6,g^b=g^2135 conjugacy classes — view the full character table.
| maximal subgroup of | #72249139, #72249140, #72249141, … |
| maximal normal subgroup of | #72249139, #72249140, #72249141, … |
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, 3, 3, 3, 3, 3, 7]);;
gap> SetPower(coll, 3, F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);