Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C24×D8 ⋊ D3 |
4181 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,3,2)(6,8);(3,4)(5,6)(7,8)(10,14)(13,15);(10,13)(11,16)(14,15);(9,10,11,15,12,14,16,13)Polycyclic presentation:
2^7,3,2^2|a^2=e,e^2=g,b^a=be,c^a=cg,d^c=df,e^b=eg,h^c=h^2,i^h=j,j^c=ij,j^h=ij86 conjugacy classes — view the full character table.
| maximal subgroup of | #72258808, AGL(1,ℤ/8)×GL(2,ℤ/4) |
| maximal normal subgroup of | AGL(1,ℤ/8)×GL(2,ℤ/4) |
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, 2, 2, 3, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 9, 8, F.10);;
gap> SetConjugate(coll, 10, 3, F.9*F.10);;
gap> SetConjugate(coll, 10, 8, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);