Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
2997 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 2997 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3,4)(6,16,20,11,10,18)(7,14,12,13,19,15)(8,9,17);(5,15,20,16)(6,14,13,8)(7,17,12,19)(9,11,18,10)Polycyclic presentation:
2^4,3,2^6|a^2=d,b^2=c,f^2=j,g^2=k,b^a=bd,e^a=e^2,f^a=g,f^b=fjk,f^d=fj,f^e=gj,g^a=fj,g^b=gj,g^d=gk,g^e=fg,h^e=i,h^f=hk,h^g=hj,i^a=hi,i^e=hi,i^f=ij,i^g=ijk,j^a=k,j^e=k,k^a=j,k^e=jk66 conjugacy classes — view the full character table.
| maximal subgroup of | C4×AGL(2,ℤ/4), #72243780 |
| maximal normal subgroup of | C4×AGL(2,ℤ/4) |
| maximal quotient of | #72243780 |
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, 1, F.4);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.7*F.10);;
gap> SetConjugate(coll, 7, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 7, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 8, 6, F.8*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.8*F.9);;
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*F.11);;
gap> SetConjugate(coll, 10, 1, F.11);;
gap> SetConjugate(coll, 10, 5, F.11);;
gap> SetConjugate(coll, 11, 1, F.10);;
gap> SetConjugate(coll, 11, 5, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);