Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #20579 ⋊ D3 |
2323 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 2323 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 24 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,6)(5,8)(10,15,14)(11,16,12);(1,2,3,5,4,6,7,8)(9,10)(11,13)(12,15)(14,16);(2,5)(3,7)(6,8)(11,12)(13,15)Polycyclic presentation:
2^5,3,2^4|b^2=d,d^2=e,b^a=bd,c^b=ce,d^a=de,f^a=f^2,g^a=gh,g^b=gi,g^f=h,h^b=hij,h^f=gh,i^a=j,i^f=ij,j^a=i,j^f=i52 conjugacy classes — view the full character table.
| maximal subgroup of | #72258027, #72258556, C2×#72252250 |
| maximal normal subgroup of | #72258027, C2×#72252250 |
| maximal quotient of | C2×#72252250, #72342055 |
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, 2, 3, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.10);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 10, 1, F.9);;
gap> SetConjugate(coll, 10, 6, F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);