Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
5195 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 5195 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 24 points:
(1,6,5,8,3,2,10,14)(4,23,15,11,9,21,13,18)(7,22)(12,20)(16,24)(17,19);(1,4,12)(2,7,18)(3,9,17)(5,15,22)(6,16,11)(8,19,21,14,20,23)(10,13,24)Polycyclic presentation:
2^4,3,2^6|b^2=c,f^2=j,g^2=jk,b^a=bd,e^a=ek,e^b=e^2j,e^c=ej,e^d=ej,f^a=fh,f^b=gk,f^d=fj,f^e=fg,g^a=gi,g^b=fk,g^d=gjk,g^e=fjk,h^b=ijk,h^e=hi,i^b=hj,i^e=h,j^b=jk,j^e=k,k^e=jk60 conjugacy classes — view the full character table.
| maximal quotient of | #72238104 |
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, 3, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 5, 2, F.5^2*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.7*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 2, F.6*F.11);;
gap> SetConjugate(coll, 7, 4, F.7*F.10*F.11);;
gap> SetConjugate(coll, 7, 5, F.6*F.10*F.11);;
gap> SetConjugate(coll, 8, 2, F.9*F.10*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.8*F.10);;
gap> SetConjugate(coll, 9, 5, F.8);;
gap> SetConjugate(coll, 10, 2, F.10*F.11);;
gap> SetConjugate(coll, 10, 5, F.11);;
gap> SetConjugate(coll, 11, 5, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);