Finite Group DatabasebetaFinite Group Databasebeta| order | 8748 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Fitting, Sylow | #14552206 ⋊ C22 |
1370 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 1370 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 27 points:
(2,4)(3,6)(5,7)(8,9)(11,14,18)(13,16,17)(19,25,26,22,23,27,20,21,24);(1,2,8)(3,5,7)(4,6,9)(10,14,16,12,18,13,15,11,17)(19,20)(21,26,23,27,25,24)Polycyclic presentation:
2^2,3^7|c^3=e,g^3=i,c^a=ci^2,d^c=de^2,f^b=f^2,f^c=fh,g^a=g^2i^2,g^c=gi,g^d=gi,h^b=h^2,i^a=i^2156 conjugacy classes — view the full character table.
| maximal subgroup of | #72254055 |
| maximal normal subgroup of | #72254055 |
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, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.9^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.9^2);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);