Finite Group DatabasebetaFinite Group Databasebeta| order | 9408 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C72×#120 ⋊ C2×C3 |
| Hall | C7×#89804 ⋊ C3 |
| Hall | #3862 ⋊ B(4,𝔽2) |
| Sylow | C72 ⋊ C3×B(4,𝔽2) |
857 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 857 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
Polycyclic presentation:
2^3,3,2^3,7^2|a^2=g,c^a=cf,c^b=ce,e^a=eg,f^b=fg,h^a=h^6,h^d=h^2,i^a=h^4i,i^d=h^4i^4178 conjugacy classes — view the full character table.
| maximal subgroup of | #72246153 |
| maximal normal subgroup of | #72246153 |
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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 7, 7]);;
gap> SetPower(coll, 1, F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 8, 4, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.8^4*F.9);;
gap> SetConjugate(coll, 9, 4, F.8^4*F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);