Finite Group DatabasebetaFinite Group Databasebeta| order | 14112 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C22×C3×C4 |
No aliases for this group.
| Fitting | C2×C72×D4 ⋊ C2×C32 |
| Hall | #13096573 ⋊ C32 |
| Hall | ΓL(1,𝔽8)2 ⋊ #127 |
| Sylow | C72 ⋊ C32×#127 |
1512 conjugacy classes of subgroups; only the 211 normal subgroups are shown.
This group has 1512 conjugacy classes of subgroups — too many to draw. Only its 211 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 336 |
Polycyclic presentation:
2^3,3^2,2^2,7^2|a^2=f,c^2=g,b^a=bf,c^b=cg,h^a=h^6,h^d=h^4,h^e=h^2,i^a=i^6,i^d=h^5i,i^e=i^2206 conjugacy classes — view the full character table.
| maximal subgroup of | #127×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72246400, #72246403, … |
| maximal normal subgroup of | #127×ΓL(1,𝔽8)⋊AGL(1,𝔽7), #72246400, #72246403, … |
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, 3, 2, 2, 7, 7]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 3, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 8, 4, F.8^4);;
gap> SetConjugate(coll, 8, 5, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^6);;
gap> SetConjugate(coll, 9, 4, F.8^5*F.9);;
gap> SetConjugate(coll, 9, 5, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);