Finite Group DatabasebetaFinite Group Databasebeta| order | 4704 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C72×M16 ⋊ C2×C3 |
| Hall | #13096143 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8) ⋊ #104 |
| Sylow | C72 ⋊ C3×#104 |
220 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 52 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:
2^2,3,2^3,7^2|b^2=d,e^2=f,b^a=be,d^a=df,e^a=ef,g^a=g^6,g^b=g^6,g^c=h,h^a=h^6,h^b=h^6,h^c=g^5h^3202 conjugacy classes — view the full character table.
| maximal subgroup of | #104×ΓL(1,𝔽8)⋊D7, #72246291, #72246298, … |
| maximal normal subgroup of | #104×ΓL(1,𝔽8)⋊D7, #72246291, #72246298, … |
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, 2, 3, 2, 2, 2, 7, 7]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 7, 2, F.7^6);;
gap> SetConjugate(coll, 7, 3, F.8);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 8, 2, F.8^6);;
gap> SetConjugate(coll, 8, 3, F.7^5*F.8^3);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);