Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting, derived | C42×C7 ⋊ C2×C32 |
| Hall | #6439 ⋊ C32 |
| Sylow | C7 ⋊ C3×#126⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C42×C7 | C2×C32 |
| Frattini | C22 | A4×AGL(1,𝔽7) |
| socle | C22×C7 | C2×C3×A4 |
| LCS | G ▷ C42×C7 | |
| UCS | C1 | |
104 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 104 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2,3^2,2^2,7,2^2|d^2=h,e^2=g,d^a=dgh,d^c=eh,e^a=eh,e^c=de,f^a=f^6,f^b=f^4,g^c=gh,h^c=g38 conjugacy classes — view the full character table.
| maximal subgroup of | #72260001, AGL(1,𝔽7)×#126⋊C3 |
| maximal normal subgroup of | AGL(1,𝔽7)×#126⋊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", "h");;
gap> coll := SingleCollector(F, [2, 3, 3, 2, 2, 7, 2, 2]);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8);;
gap> SetConjugate(coll, 4, 3, F.5*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^6);;
gap> SetConjugate(coll, 6, 2, F.6^4);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 3, F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);