Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Hall | #11729423 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #1889 |
| Sylow | C7 ⋊ C3×#1889 |
1156 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 1156 conjugacy classes of subgroups — too many to draw. Only its 153 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 3 generators on 19 points:
(5,6,11)(7,8,10)(14,19);(12,14,15,19)(13,16,17,18);(1,2,3,4)(6,7)(8,10)(9,11)(12,13)(14,18,19,16)(15,17)Polycyclic presentation:
2^3,3,2^4,7|a^2=f,b^2=h,b^a=be,c^a=cg,c^b=ceh,e^a=eh,f^b=fh,g^b=gh,i^a=i^6,i^d=i^4124 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#1889, #72276807, #72276984 |
| maximal normal subgroup of | AGL(1,𝔽7)×#1889, #72276807, #72276984 |
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.6);;
gap> SetPower(coll, 2, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^6);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);