Finite Group DatabasebetaFinite Group Databasebeta| order | 4704 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C72×D4 ⋊ C2×C3 |
| Hall | #13096577 ⋊ C3 |
| Hall | #3861 ⋊ AGL(1,ℤ/8) |
| Sylow | C72 ⋊ C3×AGL(1,ℤ/8) |
680 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 680 conjugacy classes of subgroups — too many to draw. Only its 121 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 392 |
Polycyclic presentation:
2^3,3,2^2,7^2|a^2=e,e^2=f,b^a=bef,c^a=cf,e^b=ef,g^a=g^6,g^d=g^2,h^a=h^6,h^d=h^2113 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/8)×#64424, #72245395, #72246239, … |
| maximal normal subgroup of | AGL(1,ℤ/8)×#64424, #72245395, #72246239, … |
| maximal quotient of | C2×#72245161, #72260102 |
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, 7, 7]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 8, 4, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);