Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
15340 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 15340 conjugacy classes of subgroups — too many to draw. Only its 63 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2,3,4)(5,17,8,6,9,12,18,10)(7,21)(11,25,20,14)(13,15,22,26)(16,28)(19,24,27,23);(5,7,14,9,16,23)(6,19,13)(8,15,25,18,26,24)(10,11,22)(12,27,28)(17,20,21)Polycyclic presentation:
2^3,3,2^7|b^2=c,e^2=i,f^2=j,b^a=bijk,d^a=dik,d^b=d^2,e^b=fi,e^d=fik,f^b=ej,f^d=efk,f^e=fk,g^b=h,g^d=h,h^b=g,h^d=gh,i^b=j,i^d=j,j^b=i,j^d=ijk112 conjugacy classes — view the full character table.
| maximal subgroup of | #72260440, #72262094 |
| maximal normal subgroup of | #72260440 |
| maximal quotient of | #72262094 |
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, 2, F.3);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.9*F.11);;
gap> SetConjugate(coll, 4, 2, F.4^2);;
gap> SetConjugate(coll, 5, 2, F.6*F.9);;
gap> SetConjugate(coll, 5, 4, F.6*F.9*F.11);;
gap> SetConjugate(coll, 6, 2, F.5*F.10);;
gap> SetConjugate(coll, 6, 4, F.5*F.6*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 7, 4, F.8);;
gap> SetConjugate(coll, 8, 2, F.7);;
gap> SetConjugate(coll, 8, 4, F.7*F.8);;
gap> SetConjugate(coll, 9, 2, F.10);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 10, 2, F.9);;
gap> SetConjugate(coll, 10, 4, F.9*F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);