Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C7×#534 ⋊ C2×C3 |
| Hall | #11740820 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #3026 |
| Sylow | C7 ⋊ C3×#3026 |
1600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,7)(3,4,6)(10,17)(13,20)(14,21)(16,22);(2,3)(4,6)(5,7)(8,11)(9,14)(10,16)(12,18)(13,19)(15,21)(17,22)(20,23);(8,10)(9,13)(11,16)(12,17)(14,19)(15,20)(18,22)(21,23);(8,9)(10,13)(11,14)(12,15)(16,19)(17,20)(18,21)(22,23)Polycyclic presentation:
2^4,3,2^3,7|d^a=dg,d^b=dh,d^c=df,f^a=fh,g^c=gh,i^c=i^6,i^e=i^4109 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 | #72277927 |
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> 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, 3, F.9^6);;
gap> SetConjugate(coll, 9, 5, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);