Finite Group DatabasebetaFinite Group Databasebeta| order | 3888 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C24 ⋊ #6875 |
207 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,3,11,5,13,24,18,23,22,20,17,27,4,14,26,16,10,7,6,9,21,19,12,25,15,2,8)(29,31,30)(32,35,34);(1,2,7,4,9,21,15,14,25,20,23,8,19,3,11,18,12,24,6,10,22,5,17,27,16,13,26)(28,31)(29,30)(33,35,34)Polycyclic presentation:
3^5,2^4|a^3=c,b^3=d,c^3=e^2,d^3=e^2,b^a=bd,c^b=ce,d^a=de^2,f^a=g,f^b=fg,g^a=fg,g^b=f,h^a=fgh,h^b=fi,i^a=fi,i^b=ghi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72229286, C2×#72229244 |
| maximal normal subgroup of | #72229286, C2×#72229244 |
| maximal quotient of | C2×#72229244, #72276155 |
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, [3, 3, 3, 3, 3, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.5^2);;
gap> SetPower(coll, 4, F.5^2);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 1, F.4*F.5^2);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.6*F.7);;
gap> SetConjugate(coll, 7, 2, F.6);;
gap> SetConjugate(coll, 8, 1, F.6*F.7*F.8);;
gap> SetConjugate(coll, 8, 2, F.6*F.9);;
gap> SetConjugate(coll, 9, 1, F.6*F.9);;
gap> SetConjugate(coll, 9, 2, F.7*F.8*F.9);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);