Finite Group DatabasebetaFinite Group Databasebeta| order | 3888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×C9×3−1+2 ⋊ C22 |
| Sylow | C9×3−1+2 ⋊ C2×D4 |
727 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 727 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 22 points:
(6,8)(7,11)(9,12)(10,13)(14,21,18,15,17,19)(16,20);(2,4)(5,12,13,7,6,8,11,10,9);(1,2)(3,4)(15,16)(17,18)(19,22)(20,21)Polycyclic presentation:
2^3,3,2,3^4|f^3=h,g^3=i,b^a=be,f^a=f^2h^2,f^c=f^2h^2,f^d=fh,g^a=f^2g,g^c=g^2i^2,g^d=gh^2,h^a=h^2,h^c=h^2,i^a=h^2i,i^c=i^2174 conjugacy classes — view the full character table.
| maximal subgroup of | #72228965 |
| maximal normal subgroup of | #72228965 |
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, 3, 3, 3, 3]);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.8^2);;
gap> SetConjugate(coll, 6, 3, F.6^2*F.8^2);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.6^2*F.7);;
gap> SetConjugate(coll, 7, 3, F.7^2*F.9^2);;
gap> SetConjugate(coll, 7, 4, F.7*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 3, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);