Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C34 · Schur multiplier (H2) = C35 |
No aliases for this group.
| Fitting, Sylow | #14555345 ⋊ C2 |
3049 conjugacy classes of subgroups; only the 668 normal subgroups are shown.
This group has 3049 conjugacy classes of subgroups — too many to draw. Only its 668 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) = 54 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(3,6)(5,8)(7,9)(10,17,29,20,19,18,13,12,28)(11,26,31,21,23,27,14,16,34)(15,32,35,24,30,33,22,25,36);(1,3,8,4,7,9,2,5,6)(10,14,24,13,21,15,20,11,22)(12,23,32,19,26,25,17,16,30)(18,31,36,29,34,33,28,27,35);(3,7,5)(6,9,8)(11,14,21)(15,24,22)(16,23,26)(25,32,30)(27,31,34)(33,36,35);(12,17,19)(16,26,23)(18,29,28)(25,32,30)(27,31,34)(33,35,36)Polycyclic presentation:
2,3^7|d^3=f,e^3=f^2,g^3=h,d^a=dh^2,d^c=df,e^b=ef^2,g^a=g^2h^2,g^c=gh,h^a=h^2282 conjugacy classes — view the full character table.
| maximal subgroup of | #72243364 |
| maximal normal subgroup of | #72243364 |
| maximal quotient of | 3−1+2×#92265, #72243574, #72243853 |
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");;
gap> coll := SingleCollector(F, [2, 3, 3, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 5, F.6^2);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 7, 3, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);