Finite Group DatabasebetaFinite Group Databasebeta| order | 2688 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| Fitting | C22×C42×C7 ⋊ C2×C3 |
| Hall | #11729291 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #1437 |
| Sylow | C7 ⋊ C3×#1437 |
1656 conjugacy classes of subgroups; only the 381 normal subgroups are shown.
This group has 1656 conjugacy classes of subgroups — too many to draw. Only its 381 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) = 448 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,7)(3,4,6)(12,13,14,15)(21,23);(2,3)(4,6)(5,7)(17,19)(20,21)(22,23);(8,9,10,11)(16,17)(18,19)Polycyclic presentation:
2^3,3,2^4,7|a^2=f,c^2=h,b^a=be,c^b=cg,i^b=i^6,i^d=i^4232 conjugacy classes — view the full character table.
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, 3, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 9, 2, F.9^6);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);