Finite Group DatabasebetaFinite Group Databasebeta| order | 8400 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | #72243092 ⋊ C3 |
| Hall | C52×ΓL(1,𝔽8) ⋊ C4⋊C4 |
| Hall | C52×C7 ⋊ C3×C4⋊C4 |
| Sylow | C52 ⋊ (ΓL(1,𝔽8)⋊C4)⋊C4 |
| Hall | ΓL(1,𝔽8) ⋊ #87832 |
| Sylow | C7 ⋊ C3×#87832 |
272 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 140 |
Polycyclic presentation:
2^2,3,2^2,5^2,7|a^2=e,b^2=d,b^a=be,f^a=f^4,f^b=f^3,f^d=f^4,g^a=f^2g,g^b=g^3,g^d=g^4,h^a=h^6,h^b=h^6,h^c=h^486 conjugacy classes — view the full character table.
| maximal subgroup of | #72243208, AGL(1,𝔽7)×#87832, #72243227 |
| maximal normal subgroup of | #72243208, AGL(1,𝔽7)×#87832, #72243227 |
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, 5, 5, 7]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^4);;
gap> SetConjugate(coll, 6, 2, F.6^3);;
gap> SetConjugate(coll, 6, 4, F.6^4);;
gap> SetConjugate(coll, 7, 1, F.6^2*F.7);;
gap> SetConjugate(coll, 7, 2, F.7^3);;
gap> SetConjugate(coll, 7, 4, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 8, 2, F.8^6);;
gap> SetConjugate(coll, 8, 3, F.8^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);