Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C2×C35 |
No aliases for this group.
| Fitting | C22×3−1+4 ⋊ C3 |
| Sylow | C22 ⋊ #10624749 |
718 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 718 conjugacy classes of subgroups — too many to draw. Only its 70 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:
(1,4)(2,3)(5,7,14,8,15,24,16,13,23)(6,10,20,12,19,27,11,22,28)(9,17,25,18,26,31,21,29,30);(2,4,3)(7,13,15)(10,19,22)(20,27,28)(25,30,31);(5,6,9)(7,10,17)(8,12,18)(11,21,16)(13,22,29)(14,20,25)(15,19,26)(23,28,30)(24,27,31)Polycyclic presentation:
3^6,2^2|c^3=f,b^a=be,c^b=cd,d^a=df,d^c=df,e^c=ef^2,g^b=h,h^b=gh140 conjugacy classes — view the full character table.
| derived subgroup of | #72259917 |
| maximal subgroup of | A4×#10624749, #72259917 |
| maximal normal subgroup of | A4×#10624749, #72259917 |
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, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 3, F.5*F.6^2);;
gap> SetConjugate(coll, 7, 2, F.8);;
gap> SetConjugate(coll, 8, 2, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);