Finite Group DatabasebetaFinite Group Databasebeta| order | 3402 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C32 |
No aliases for this group.
| Fitting, Hall | C7×3−1+4 ⋊ C2 |
| derived | C3×C7×C9 ⋊ C2×C32 |
| Sylow | 3−1+4 ⋊ D7 |
| Sylow | C7 ⋊ #4207⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C7×C9 | C2×C32 |
| Frattini | C3 | C32×C3⋊D21 |
| Fitting | C7×3−1+4 | C2 |
| socle | C3×C7 | C32×C3⋊D3 |
| LCS | G ▷ C3×C7×C9 | |
| UCS | C1 | |
362 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 362 conjugacy classes of subgroups — too many to draw. Only its 80 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:
2,3^4,7,3|e^3=g,b^a=bg^2,d^a=d^2,d^b=dg^2,e^a=e^2g^2,e^b=eg^2,e^c=eg,f^a=f^6,g^a=g^2304 conjugacy classes — view the full character table.
| maximal subgroup of | #72249205, #72254856, D7×#4207⋊C3 |
| maximal normal subgroup of | #72254856, D7×#4207⋊C3 |
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, 7, 3]);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.7^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.7^2);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.7^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.7^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);