Finite Group DatabasebetaFinite Group Databasebeta| order | 3456 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C32×Q8 ⋊ #215 |
327 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 327 conjugacy classes of subgroups — too many to draw. Only its 26 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) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,8,4,7,6)(3,5)(9,17,14,10)(11,15,13,16);(2,5)(3,7)(6,8)(10,12)(11,16)(14,15)Polycyclic presentation:
2^4,3^3,2^3|b^2=d,c^2=d,h^2=j,i^2=j,b^a=bc,c^a=cd,c^b=cd,e^a=e^2,f^a=fg^2,f^b=g^2,f^c=f^2g,f^d=f^2,g^a=g^2,g^b=f,g^c=fg,g^d=g^2,h^a=ij,h^e=i,i^a=hj,i^e=hij,i^h=ij39 conjugacy classes — view the full character table.
| maximal subgroup of | #72244547, GL(2,𝔽3)×AΓL(1,𝔽9) |
| maximal normal subgroup of | GL(2,𝔽3)×AΓL(1,𝔽9) |
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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 3, 3, 2, 2, 2]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6*F.7^2);;
gap> SetConjugate(coll, 6, 2, F.7^2);;
gap> SetConjugate(coll, 6, 3, F.6^2*F.7);;
gap> SetConjugate(coll, 6, 4, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.6);;
gap> SetConjugate(coll, 7, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 4, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.9*F.10);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);