Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | #72282115 ⋊ C3 |
3495 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 3495 conjugacy classes of subgroups — too many to draw. Only its 110 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:
(1,3,5,18,8,16,4,17)(2,10,11,15,14,36,9,31)(6,20,53,47,30,50,21,46,29,19,24,63,27,44,54,45,7,52,23,48,26,43,22,64)(12,35,57,41,28,59,32,51,40,34,38,62,25,56,58,49,13,60,37,42,39,55,33,61);(1,26,8,29)(2,57)(3,52,16,50)(4,53,5,54)(6,30,27,7)(9,40)(10,61)(11,39)(12,38)(13,32)(14,58)(15,60)(17,64,18,63)(19,20,43,44)(21,24,23,22)(25,33)(28,37)(31,59)(34,51)(35,41)(36,62)(42,55)(45,46,47,48)(49,56);(1,2,4,11,8,14,5,9)(3,15,17,36,16,31,18,10)(6,25,21,32,27,12,23,37)(7,28,22,33,30,13,24,38)(19,49,45,34,43,41,47,55)(20,51,46,35,44,42,48,56)(26,40,53,57,29,39,54,58)(50,62,63,59,52,61,64,60)Polycyclic presentation:
2^3,3,2^7|a^2=e,b^2=ef,c^2=f,e^2=k,f^2=k,g^2=k,h^2=k,b^a=bf,c^a=cf,c^b=ck,f^a=fk,g^a=gj,g^d=gh,h^a=hi,h^d=g,h^g=hk,i^d=j,i^g=ik,j^d=ijk,j^h=jk118 conjugacy classes — view the full character table.
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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 2, F.5*F.6);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 6, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.7);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 10, 4, F.9*F.10*F.11);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);