Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C2×C35 |
No aliases for this group.
| Fitting | C22×#6909 ⋊ C3 |
| Sylow | C22 ⋊ #10624749 |
562 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 562 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 3 generators on 31 points:
(2,4,3)(5,7,15,8,17,14,18,16,28)(6,11,25,13,26,23,12,10,24)(9,20,30,21,19,29,27,22,31);(1,4)(2,3)(5,7,14)(6,10,23)(8,17,28)(9,19,29)(11,24,13)(12,26,25)(15,18,16)(20,30,27)(21,22,31);(5,6,9)(7,11,20)(8,13,21)(10,22,16)(12,27,18)(14,23,29)(15,25,30)(17,26,19)(24,31,28)Polycyclic presentation:
3^6,2^2|c^3=f,b^a=be,c^b=cd,d^a=df,d^b=df,d^c=df,e^b=ef^2,e^c=ef^2,g^c=h,h^c=gh92 conjugacy classes — view the full character table.
| derived subgroup of | #72259892 |
| maximal subgroup of | A4×#10624749, #72259892 |
| maximal normal subgroup of | A4×#10624749, #72259892 |
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, [3, 3, 3, 3, 3, 3, 2, 2]);;
gap> SetPower(coll, 3, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.6^2);;
gap> SetConjugate(coll, 7, 3, F.8);;
gap> SetConjugate(coll, 8, 3, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);