Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| Fitting | C4×C7×M16 ⋊ C2×C3 |
| Hall | #11735975 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #3151 |
| Sylow | C7 ⋊ C3×#3151 |
1312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 224 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,7)(3,4,6)(8,10,11,9)(12,13,14,16,15,18,19,17);(8,9,11,10);(2,3)(4,6)(5,7)(8,9)(10,11)(13,17)(14,19)(16,18);(2,3)(4,6)(5,7)(9,10)(13,16)(14,19)(17,18)Polycyclic presentation:
2^4,3,2^3,7|c^2=f,d^2=g,g^2=h,b^a=bf,c^a=cf,c^b=cf,d^a=dg,d^b=dgh,g^a=gh,g^b=gh,i^a=i^6,i^b=i^6,i^e=i^4136 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#3151, #72259660, #72259710 |
| maximal normal subgroup of | AGL(1,𝔽7)×#3151, #72259660, #72259710 |
| maximal quotient of | #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, 2, 3, 2, 2, 2, 7]);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
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.7*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^6);;
gap> SetConjugate(coll, 9, 2, F.9^6);;
gap> SetConjugate(coll, 9, 5, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);