Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C7×#585 ⋊ C2×C3 |
| Hall | #11741570 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #3025 |
| Sylow | C7 ⋊ C3×#3025 |
1532 conjugacy classes of subgroups; only the 267 normal subgroups are shown.
This group has 1532 conjugacy classes of subgroups — too many to draw. Only its 267 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 — 4 generators on 23 points:
(1,2,7)(3,4,6)(8,11,12,19)(9,14,15,23)(10,16,17,18)(13,21,20,22);(2,3)(4,6)(5,7)(13,20)(21,22);(8,10)(9,13)(11,18)(12,17)(14,22)(15,20)(16,19)(21,23);(8,9)(10,13)(11,14)(12,15)(16,21)(17,20)(18,22)(19,23)Polycyclic presentation:
2^4,3,2^3,7|d^2=h,c^a=cg,c^b=cf,d^b=dh,f^a=fh,g^b=gh,i^c=i^6,i^e=i^4130 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#3025 |
| maximal normal subgroup of | AGL(1,𝔽7)×#3025 |
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, 4, F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, 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);