Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting, Sylow | #72281258 ⋊ C3 |
3248 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 3248 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,7,11,10,14,15,24)(8,13,34,22,20,27,31,29,21,12,17,36,18,28,33,26,9,30,16,23,19,25,32,35);(1,2,3,4)(5,19)(6,35)(7,34)(8,9,18,20)(10,21)(11,27)(12,23,25,29)(13,26,28,22)(14,36)(15,33)(16,32,31,17)(24,30);(8,18)(9,20)(12,25)(13,28)(16,31)(17,32)(19,21)(22,26)(23,29)(27,30)(33,34)(35,36)Polycyclic presentation:
2^3,3,2^7|b^2=e,c^2=f,f^2=k,g^2=k,h^2=k,c^b=cf,f^b=fk,g^a=gj,g^b=gj,g^d=gh,h^a=hi,h^b=hi,h^d=g,h^g=hk,i^d=j,i^g=ik,j^d=ijk,j^h=jk124 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, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.5);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 6, F.11);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 8, F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.7);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 9, 4, F.10);;
gap> SetConjugate(coll, 9, 7, F.9*F.11);;
gap> SetConjugate(coll, 10, 4, F.9*F.10*F.11);;
gap> SetConjugate(coll, 10, 8, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);