Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C2×C35 |
No aliases for this group.
| Fitting | C22×C3×AΣL(1,𝔽27) ⋊ C3 |
| Sylow | C22 ⋊ #10624749 |
614 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 614 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) = 54 |
A minimal faithful permutation representation — 3 generators on 31 points:
(2,4,3)(5,7,14,8,15,24,16,13,23)(6,11,20,10,19,28,12,21,29)(9,17,26,18,27,31,22,25,30);(1,4)(2,3)(6,10,12)(7,13,15)(17,25,27)(20,28,29);(5,6,9,16,12,22,8,10,18)(7,11,17,13,21,25,15,19,27)(14,20,26,23,29,30,24,28,31)Polycyclic presentation:
3^6,2^2|a^3=f^2,c^3=f,b^a=bd,c^b=cde^2,d^a=df,e^a=ef,e^c=ef^2,g^c=h,h^c=gh108 conjugacy classes — view the full character table.
| derived subgroup of | #72260092 |
| maximal subgroup of | A4×#10624749, #72260092 |
| maximal normal subgroup of | A4×#10624749, #72260092 |
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, 1, F.6^2);;
gap> SetPower(coll, 3, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.4);;
gap> SetConjugate(coll, 3, 2, F.3*F.4*F.5^2);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
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);