Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Hall | #11729608 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #1910 |
| Sylow | C7 ⋊ C3×#1910 |
688 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 688 conjugacy classes of subgroups — too many to draw. Only its 147 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 — 3 generators on 23 points:
(1,2,7)(3,4,6)(10,16)(11,18)(13,19)(14,22)(17,23)(20,21);(2,3)(4,6)(5,7)(8,10,11,16,12,17,18,23)(9,13,14,20,15,21,22,19);(8,9)(10,13)(11,14)(12,15)(16,20)(17,21)(18,22)(19,23)Polycyclic presentation:
2^3,3,2^4,7|b^2=e,e^2=h,c^a=cg,c^b=cef,e^c=eh,f^a=fh,g^b=gh,i^b=i^6,i^d=i^4109 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#1910 |
| maximal normal subgroup of | AGL(1,𝔽7)×#1910 |
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, 2, F.5);;
gap> SetPower(coll, 5, F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*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);