Finite Group DatabasebetaFinite Group Databasebeta| order | 3888 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
248 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 144 |
A minimal faithful permutation representation — 2 generators on 34 points:
(1,12,13,9,5,6,10,7)(2,3,16,4,8,11,15,14)(17,20,25,19,23,21,22,18,24)(26,27)(28,34)(30,31)(32,33);(3,10,4)(6,15,7)(9,12,16)(11,13,14)(18,20,23)(21,25,24)(26,32,33,29,30,34,27,28,31)Polycyclic presentation:
2,3^5,2^3|a^2=i,c^3=d,e^3=f,g^2=i,h^2=i,c^a=cf^2,c^b=cd,e^a=e^2f^2,e^b=ef,f^a=f^2,g^a=hi,g^e=gh,h^a=gi,h^e=g,h^g=hi117 conjugacy classes — view the full character table.
| maximal subgroup of | #72253023 |
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");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 3, 3, 2, 2, 2]);;
gap> SetPower(coll, 1, F.9);;
gap> SetPower(coll, 3, F.4);;
gap> SetPower(coll, 5, F.6);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.6^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.6^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.8*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 5, F.7);;
gap> SetConjugate(coll, 8, 7, F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);