Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C24×D8 ⋊ D3 |
3707 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 3707 conjugacy classes of subgroups — too many to draw. Only its 55 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) = 48 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,4,3,2)(5,8,6,7)(10,14)(13,15);(2,3,4)(6,7,8)(10,13)(11,16)(14,15);(9,10,11,15,12,14,16,13)Polycyclic presentation:
2^5,3,2^4|a^2=d,d^2=e,b^a=bd,c^a=ce,d^b=de,f^c=f^2,g^f=h,h^c=gh,h^f=gh,i^c=hij,i^f=j,j^c=gj,j^f=ij58 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/8)×#1020, #72258808 |
| maximal normal subgroup of | AGL(1,ℤ/8)×#1020 |
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.4);;
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, 2, F.4*F.5);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 10, 3, F.7*F.10);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);