Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | #6628842 ⋊ C3 |
10191 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 10191 conjugacy classes of subgroups — too many to draw. Only its 69 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:
(3,4,8)(5,6,7)(10,14)(13,16);(1,7,4,2)(3,5,8,6)(9,10)(11,16)(12,14)(13,15);(10,13)(11,15)(14,16)Polycyclic presentation:
2^3,3,2^6|e^2=f,b^a=be,c^b=cf,e^a=ef,e^b=ef,g^b=gj,g^d=gh,h^b=hi,h^d=g,i^d=j,j^d=ij64 conjugacy classes — view the full character table.
| maximal subgroup of | #72234205, #72258027, C2×#72233784 |
| maximal normal subgroup of | #72258027, C2×#72233784 |
| maximal quotient of | C2×#72233784, #72341937, #72342118 |
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, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.7);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 10, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);