Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Hall | C3×ΓL(1,𝔽8) ⋊ C2×C22⋊C4 |
| Sylow | C7 ⋊ C3×D3×C22⋊C4 |
562 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 562 conjugacy classes of subgroups — too many to draw. Only its 117 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:
2^3,3,2^2,3,7|a^2=e,c^a=cf,g^b=g^2,h^a=h^6,h^b=h^6,h^d=h^4108 conjugacy classes — view the full character table.
| maximal subgroup of | #72244193, #72244206, D3×C22⋊C4×AGL(1,𝔽7), … |
| maximal normal subgroup of | #72244193, #72244206, D3×C22⋊C4×AGL(1,𝔽7), … |
| maximal quotient of | #72244555, #72247206, #72254344, … |
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, 2, 3, 2, 2, 3, 7]);;
gap> SetPower(coll, 1, F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 8, 2, F.8^6);;
gap> SetConjugate(coll, 8, 4, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);