Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | #63076 ⋊ D3 |
2753 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 2753 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 3 generators on 64 points:
(3,14)(5,29,30)(6,26,25)(8,31)(10,40,23)(11,27,38)(13,34)(15,19)(16,46,24,28,48,17)(20,55,56)(21,54,53)(32,57,37,39,58,33)(35,60,50)(36,49,59)(41,61,45,47,62,42)(43,63,52,51,64,44);(1,28,25,17,7,16,29,24)(2,39,38,33,12,32,40,37)(3,26,46,6,14,5,48,30)(4,43,53,52,22,51,55,44)(8,27,57,11,31,10,58,23)(9,41,59,45,18,47,60,42)(13,35,61,50,34,49,62,36)(15,20,63,56,19,54,64,21);(1,2,7,12)(3,13,14,34)(4,18,22,9)(5,23,26,11)(6,27,30,10)(8,19,31,15)(16,45,28,42)(17,47,24,41)(20,36,54,50)(21,35,56,49)(25,40,29,38)(32,44,39,52)(33,43,37,51)(46,62,48,61)(53,59,55,60)(57,63,58,64)Polycyclic presentation:
2^4,3,2^5|a^2=j,d^2=j,f^2=j,g^2=j,b^a=bd,c^b=cj,d^a=dj,d^b=dj,e^a=e^2,e^b=e^2,f^a=gj,f^b=gj,f^e=fg,g^a=fj,g^b=fj,g^e=f,g^f=gj,h^a=hi,h^b=hi,h^e=i,h^g=hj,i^e=hi,i^f=ij,i^g=ij59 conjugacy classes — view the full character table.
| maximal subgroup of | #72341786 |
| maximal normal subgroup of | #72341786 |
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, 1, F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.7*F.10);;
gap> SetConjugate(coll, 6, 2, F.7*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 5, F.6);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 7, 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);