Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
1241 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 1241 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,3,4)(5,10,14)(6,9,8)(7,11,16,13,12,15)(17,19,20,24)(18,21,22,23);(2,3)(5,13,8,7)(6,15,12,9,10,16,11,14)(17,18,20,22)(19,23,24,21)Polycyclic presentation:
2^3,3,2^6|a^2=c,b^2=cj,e^2=i,f^2=j,b^a=bcj,d^a=d^2i,d^c=di,e^a=efi,e^b=ei,e^d=fi,f^b=fj,f^d=ef,g^a=h,g^d=gh,h^a=g,h^d=g,i^a=ij,i^d=j,j^d=ij48 conjugacy classes — view the full character table.
| maximal subgroup of | #72267875 |
| maximal normal subgroup of | #72267875 |
| maximal quotient of | #72267879 |
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, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.3*F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 4, F.7);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);