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×H(3,𝔽3) ⋊ C2×C3 |
| Hall | #13133630 ⋊ C2 |
| Hall | D7 ⋊ #6909 |
| Sylow | C7 ⋊ C2×#6909 |
296 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 77 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|b^3=f^2,c^b=ce,d^b=df,d^c=df^2,e^b=ef,g^a=g^6,g^b=g^2103 conjugacy classes — view the full character table.
| maximal subgroup of | #72249123, AGL(1,𝔽7)×#6909, #72249139, … |
| maximal normal subgroup of | #72249123, AGL(1,𝔽7)×#6909, #72249139, … |
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, 2, F.6^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.6^2);;
gap> SetConjugate(coll, 5, 2, F.5*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);