Finite Group DatabasebetaFinite Group Databasebeta| order | 4608 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
3563 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 3563 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,4,2,3)(5,7,8,6);(2,3,4)(10,13)(11,15)(14,16);(7,8)(9,10,11,13,12,14,15,16)Polycyclic presentation:
2^5,3^2,2^4|a^2=d,d^2=e,b^a=bde,d^b=de,f^a=f^2,f^c=f^2,g^a=f^2g,g^c=g^2,h^f=i,h^g=hi,i^a=hi,i^c=hi,i^f=hi,i^g=h,j^a=hj,j^c=hj,j^f=hj,j^g=k,k^c=ijk,k^f=hik,k^g=jk89 conjugacy classes — view the full character table.
| maximal subgroup of | #72258669 |
| maximal normal subgroup of | #72258669 |
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.6^2*F.7);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 8, 6, F.9);;
gap> SetConjugate(coll, 8, 7, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
gap> SetConjugate(coll, 9, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.8*F.9);;
gap> SetConjugate(coll, 9, 7, F.8);;
gap> SetConjugate(coll, 10, 1, F.8*F.10);;
gap> SetConjugate(coll, 10, 3, F.8*F.10);;
gap> SetConjugate(coll, 10, 6, F.8*F.10);;
gap> SetConjugate(coll, 10, 7, F.11);;
gap> SetConjugate(coll, 11, 3, F.9*F.10*F.11);;
gap> SetConjugate(coll, 11, 6, F.8*F.9*F.11);;
gap> SetConjugate(coll, 11, 7, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);