Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C8 |
No aliases for this group.
| Sylow | C3 ⋊ #212918 |
2452 conjugacy classes of subgroups; only the 461 normal subgroups are shown.
This group has 2452 conjugacy classes of subgroups — too many to draw. Only its 461 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 768 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3)(16,17,18,20,19,21,22,23);(8,9,10,12,11,13,14,15);(2,3)(4,6,7,5)(9,13)(12,15)(17,21)(20,23);(2,3)(4,5,7,6)(9,12)(10,14)(13,15)(17,20)(18,22)(21,23)Polycyclic presentation:
2^9,3|a^2=e,b^2=e,c^2=f,d^2=g,f^2=h,g^2=i,c^a=cf,c^b=ch,d^a=dg,d^b=di,f^a=fh,g^a=gi,j^a=j^2,j^b=j^2156 conjugacy classes — view the full character table.
| maximal subgroup of | #72225906, D3×#212918 |
| maximal normal subgroup of | #72225906, D3×#212918 |
| maximal quotient of | #72225906 |
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.5);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 10, 1, F.10^2);;
gap> SetConjugate(coll, 10, 2, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);