Finite Group DatabasebetaFinite Group Databasebeta| order | 2592 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| derived | C9×A4 ⋊ C2×C3×C4 |
421 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 421 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,3,4,2)(5,8,7)(10,14,15,12,13,17)(11,16);(1,2,4,3)(6,8)(9,15,12,11,13,14,16,10,17)Polycyclic presentation:
2^2,3,2,3^3,2^2|a^2=d,b^2=d,f^3=g,e^a=e^2,f^b=f^2g^2,f^c=fg^2,g^b=g^2,h^a=hi,h^e=i,i^e=hi100 conjugacy classes — view the full character table.
| maximal subgroup of | #72254833, C4×S4×AGL(1,ℤ/9) |
| maximal normal subgroup of | C4×S4×AGL(1,ℤ/9) |
| maximal quotient of | #72254833 |
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, 2, 3, 3, 3, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 2, F.6^2*F.7^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.7^2);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);