Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C16 |
No aliases for this group.
| Sylow | C3 ⋊ #123419 |
520 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 133 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3)(20,21,27,35,23,28,22,25,24,29,30,34,31,33,32,26);(5,9)(6,13)(7,14)(10,11)(12,18)(15,19)(16,17)(21,26)(22,30)(23,31)(25,29)(27,32)(28,34)(33,35);(2,3)(4,5,8,12)(6,9,15,18)(7,16,14,11)(10,19,17,13)(21,25)(22,30)(23,31)(26,29)(27,32)(28,35)(33,34)Polycyclic presentation:
2^9,3|a^2=h,c^2=egi,d^2=fh,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,c^b=cei,d^a=df,d^b=df,e^a=eg,e^b=eg,f^a=fh,f^b=fh,g^a=gi,g^b=gi,j^a=j^2156 conjugacy classes — view the full character table.
| maximal subgroup of | #72226167 |
| maximal normal subgroup of | #72226167 |
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, 2, 2, 3]);;
gap> SetPower(coll, 1, F.8);;
gap> SetPower(coll, 3, F.5*F.7*F.9);;
gap> SetPower(coll, 4, F.6*F.8);;
gap> SetPower(coll, 5, F.7*F.9);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.7*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);