Finite Group DatabasebetaFinite Group Databasebeta| order | 13122 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C36 |
No aliases for this group.
| Fitting, Sylow | #72243615 ⋊ C2 |
| derived | C33×C9 ⋊ C2×C33 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C33×C9 | C2×C33 |
| Frattini | C33 | C33×C3⋊D3 |
| Fitting | #72243615 | C2 |
| socle | C3 | C3×#13091503 |
| LCS | G ▷ C33×C9 | |
| UCS | C1 | |
8607 conjugacy classes of subgroups; only the 251 normal subgroups are shown.
This group has 8607 conjugacy classes of subgroups — too many to draw. Only its 251 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,3,11)(2,7,23)(4,35,37,14,12,64)(5,39,34,18,13,33)(6,19,43)(8,25,56,10,26,51)(9,55,57,28,30,76)(15,63,62,40,36,61)(16,53,52,41,24,75)(17,65,60,42,38,32)(20,73,67,47,49,79)(21,44,66,22,46,68)(27,59,74,31,54,77)(29,70,71,50,45,80)(48,72,78,58,69,81);(1,16,58)(2,48,14)(3,54,72)(4,28,22)(5,27,47)(6,15,31)(7,49,63)(8,20,40)(9,50,18)(10,29,17)(11,74,79)(12,59,44)(13,30,73)(19,36,53)(21,42,41)(23,78,61)(24,70,39)(25,69,35)(26,45,38)(32,76,68)(33,52,81)(34,57,80)(37,77,43)(46,65,55)(51,71,64)(56,67,60)(62,75,66);(2,8,10)(4,15,42)(12,36,65)(14,40,17)(19,44,46)(20,48,29)(23,51,56)(24,54,30)(28,31,41)(32,37,62)(35,63,38)(43,66,68)(47,58,50)(52,57,74)(53,59,55)(60,64,61)(67,71,78)(70,72,73)Polycyclic presentation:
2,3^8|f^3=i,e^a=e^2,e^c=eg,e^d=eh,f^a=f^2i^2,f^b=fi,f^c=fg,f^d=fg^2i^2,g^a=g^2,g^c=gi^2,g^d=gi,h^a=h^2,h^c=hi,i^a=i^2225 conjugacy classes — view the full character table.
| maximal subgroup of | #72243793, #72243956 |
| maximal normal subgroup of | #72243793, #72243956 |
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, 3, 3, 3]);;
gap> SetPower(coll, 6, F.9);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.9^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.7^2*F.9^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7*F.9^2);;
gap> SetConjugate(coll, 7, 4, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);