Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
2388 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 2388 conjugacy classes of subgroups — too many to draw. Only its 70 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:
(1,2,3,8,4,10,13,24,7,9,16,23,17,27,36,30)(5,29,37,25,41,54,34,53,43,28,15,60,40,32,64,52,22,33,14,45,42,31,57,61,21,12,58,44,19,62,56,47,20,11,39,48,18,50,55,46,6,51,38,26,59,49,35,63);(1,5,20,6)(2,25,51,45)(3,34,37,35)(4,40,42,41)(7,21,43,22)(8,28,60,12)(9,44,33,26)(10,52,62,47)(11,48,29,23)(13,14,55,15)(16,56,58,57)(17,18,59,19)(24,31,63,50)(27,46,54,53)(30,49,61,32)(36,38,64,39)Polycyclic presentation:
2^5,3,2^5|a^2=c,b^2=d,d^2=e,e^2=k,b^a=bd,c^b=ck,d^a=de,e^a=ek,f^a=f^2,g^a=h,g^f=h,h^a=g,h^f=gh,i^a=ij,i^f=j,i^g=ik,j^f=ij,j^g=jk,j^h=jk94 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, 2, 2, 3, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 5, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.8);;
gap> SetConjugate(coll, 7, 6, F.8);;
gap> SetConjugate(coll, 8, 1, F.7);;
gap> SetConjugate(coll, 8, 6, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 6, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 10, 6, F.9*F.10);;
gap> SetConjugate(coll, 10, 7, F.10*F.11);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);