Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | #62624 ⋊ D3 |
672 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 672 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,9,29,4,10,3,8,7,13,14,25,15,24,18,28)(5,36,55,53,40,56,16,52,45,35,31,62,39,33,60,51,23,34,30,49,44,32,43,50,22,12,58,48,20,61,42,27,21,11,38,64,19,47,41,26,6,57,37,54,59,46,17,63);(1,5,21,6)(2,26,34,27)(3,37,41,38)(4,39,44,40)(7,22,45,23)(8,11,50,12)(9,42,55,43)(10,53,56,54)(13,51,57,52)(14,16,58,17)(15,19,59,20)(18,30,60,31)(24,48,61,49)(25,46,62,47)(28,35,63,36)(29,32,64,33)Polycyclic presentation:
2^4,3,2^5|b^2=cdj,c^2=dj,d^2=j,b^a=bc,c^a=cd,d^a=dj,e^a=e^2,f^a=g,f^e=g,g^a=f,g^e=fg,h^a=hi,h^e=i,h^f=hj,i^e=hi,i^f=ij,i^g=ij56 conjugacy classes — view the full character table.
| maximal subgroup of | #72342007 |
| maximal normal subgroup of | #72342007 |
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, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3*F.4*F.10);;
gap> SetPower(coll, 3, F.4*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 1, F.6);;
gap> SetConjugate(coll, 7, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 6, F.9*F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);