Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #36614 ⋊ D3 |
2065 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 2065 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:
(2,8,6)(3,4,7)(10,14)(13,15);(3,7)(5,8)(10,13)(11,16)(14,15);(1,2)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,16,15)Polycyclic presentation:
2^5,3,2^4|a^2=e,d^2=e,b^a=bd,c^a=ce,d^a=de,d^b=de,f^b=f^2,g^a=gi,g^b=gh,g^f=h,h^a=hj,h^f=gh,i^b=ij,i^f=j,j^f=ij52 conjugacy classes — view the full character table.
| maximal subgroup of | #72258027, #72258548, C2×#72258012 |
| maximal normal subgroup of | #72258027, C2×#72258012 |
| maximal quotient of | C2×#72258012, #72342017, #72342127 |
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, 1, F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);