Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C7×#458 ⋊ C2×C3 |
| Hall | #11737756 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #3026 |
| Sylow | C7 ⋊ C3×#3026 |
1600 conjugacy classes of subgroups; only the 239 normal subgroups are shown.
This group has 1600 conjugacy classes of subgroups — too many to draw. Only its 239 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 4 generators on 23 points:
(2,6,5,3,4,7)(10,17)(13,20)(14,22)(16,23);(1,2,5,6,4,7,3)(8,11)(9,14)(10,16)(12,18)(13,19)(15,22)(17,23)(20,21);(8,10,12,17)(9,13,15,20)(11,16,18,23)(14,21,22,19);(8,9)(10,13)(11,14)(12,15)(16,19)(17,20)(18,22)(21,23)Polycyclic presentation:
2^4,3,2^3,7|b^2=h,b^a=bg,c^b=cf,d^a=dg,d^b=dh,d^c=df,f^a=fh,g^c=gh,i^d=i^6,i^e=i^4103 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#3026 |
| maximal normal subgroup of | AGL(1,𝔽7)×#3026 |
| maximal quotient of | #72277311 |
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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 2, 2, 2, 7]);;
gap> SetPower(coll, 2, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 9, 4, F.9^6);;
gap> SetConjugate(coll, 9, 5, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);