Finite Group DatabasebetaFinite Group Databasebeta| order | 3072 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting, Sylow | #72340689 ⋊ C3 |
4296 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 4296 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,9,38,7,3,23,4,18,12,11,13,35,6,25,29,20,15,28,8,37,31,2,21,44)(5,24,45,19,14,42,16,36,34,27,32,48,17,43,46,26,33,40,22,41,30,10,39,47);(2,7)(4,8)(5,17)(9,18)(10,26)(11,20)(13,21)(19,27)(23,35)(24,36)(25,37)(28,44)(29,38)(32,39)(40,47)(41,43)(42,48)(45,46);(1,5,6,17)(2,10,11,27)(3,14)(4,16,8,22)(7,19,20,26)(9,24)(13,32)(15,33)(18,36)(21,39)(25,43)(30,34)(37,41)(40,42)(45,46)(47,48)Polycyclic presentation:
2^2,3,2^8|b^2=dj,d^2=j,e^2=i,f^2=ik,b^a=bd,c^b=cijk,d^a=dj,e^a=ek,e^c=fij,f^a=fij,f^c=efj,f^e=fj,g^c=gh,g^e=gik,g^f=gij,h^c=g,h^e=hk,h^f=hik,i^c=ik,k^c=ij59 conjugacy classes — view the full character table.
| maximal subgroup of | #72340724, #72340745 |
| maximal normal subgroup of | #72340724, #72340745 |
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, 3, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.4*F.10);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.9*F.11);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.10*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.11);;
gap> SetConjugate(coll, 5, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.5*F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 3, F.7);;
gap> SetConjugate(coll, 8, 5, F.8*F.11);;
gap> SetConjugate(coll, 8, 6, F.8*F.9*F.11);;
gap> SetConjugate(coll, 9, 3, F.9*F.11);;
gap> SetConjugate(coll, 11, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);