Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C8 |
No aliases for this group.
| Fitting | C7×#322 ⋊ C2×C3 |
| Hall | #11725845 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #1715 |
| Sylow | C7 ⋊ C3×#1715 |
440 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 109 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) = 448 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,7)(3,4,6)(16,17,22,23,19,20,18,21);(2,3)(4,6)(5,7)(9,12)(10,14)(13,15)(17,21)(18,22)(20,23);(8,9,10,12,11,13,14,15)(17,20)(21,23)Polycyclic presentation:
2^3,3,2^4,7|a^2=e,c^2=fh,e^2=g,f^2=h,b^a=beg,c^a=ch,c^b=cf,e^b=eg,f^b=fh,i^b=i^6,i^d=i^4118 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#1715, #72259644, #72259710 |
| maximal normal subgroup of | AGL(1,𝔽7)×#1715, #72259644, #72259710 |
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, 3, 2, 2, 2, 2, 7]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.6*F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 9, 2, F.9^6);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);