Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
1399583 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 1399583 conjugacy classes of subgroups — too many to draw. Only its 139 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) = 24 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,3,4)(6,8)(9,12,11,10)(13,15,16,14)(19,20);(5,6)(7,8)(10,11,12)(14,15,16)(18,19,20);(17,20)(18,19)Polycyclic presentation:
2^4,3,2^6|b^2=c,b^a=bd,e^b=e^2,f^e=g,g^b=fg,g^e=fg,h^b=fgi,h^e=i,i^b=gh,i^e=hi,j^b=fghjk,j^e=k,k^b=ghik,k^e=jk144 conjugacy classes — view the full character table.
| maximal subgroup of | #72226608, #72226616, #72227399 |
| maximal quotient of | #72267529, #72268761 |
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> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 6, 5, F.7);;
gap> SetConjugate(coll, 7, 2, F.6*F.7);;
gap> SetConjugate(coll, 7, 5, F.6*F.7);;
gap> SetConjugate(coll, 8, 2, F.6*F.7*F.9);;
gap> SetConjugate(coll, 8, 5, F.9);;
gap> SetConjugate(coll, 9, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 5, F.8*F.9);;
gap> SetConjugate(coll, 10, 2, F.6*F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 10, 5, F.11);;
gap> SetConjugate(coll, 11, 2, F.7*F.8*F.9*F.11);;
gap> SetConjugate(coll, 11, 5, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);