Finite Group DatabasebetaFinite Group Databasebeta| order | 2016 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C3×C7×D4 ⋊ C2×C3 |
| Hall | C3×ΓL(1,𝔽8) ⋊ #120 |
| Sylow | C7 ⋊ C3×#937 |
574 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 574 conjugacy classes of subgroups — too many to draw. Only its 96 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|b^a=be,c^b=cf,g^a=g^2,h^a=h^6,h^d=h^4102 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#937, #120×C3⋊AGL(1,𝔽7), D3×#6500⋊C3, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#937, #120×C3⋊AGL(1,𝔽7), D3×#6500⋊C3, … |
| maximal quotient of | #72254407, #72254411, #72254436, … |
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> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 8, 4, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);