Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C7×#447 ⋊ C2×C3 |
| Hall | #11737447 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #3003 |
| Sylow | C7 ⋊ C3×#3003 |
1988 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 4 generators on 19 points:
(6,10,9,7,8,11)(13,17)(16,18);(5,6,9,10,8,11,7)(13,16)(14,19)(17,18);(2,4);(1,2)(3,4)(12,13,14,18,15,17,19,16)Polycyclic presentation:
2^4,3,2^3,7|a^2=f,f^2=h,b^a=bg,c^a=cf,d^a=dh,f^c=fh,i^d=i^6,i^e=i^4115 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#3003 |
| maximal normal subgroup of | AGL(1,𝔽7)×#3003 |
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, 1, F.6);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*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);