Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, Sylow | #14552587 ⋊ C2 |
1059 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 1059 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 3 generators on 27 points:
(10,13,18,12,16,14,15,11,17)(20,22)(21,25)(23,26)(24,27);(1,4,9,3,7,5,6,2,8)(19,23,22,21,27,24,25,20,26);(2,4,7)(5,9,8)(11,13,16)(14,18,17)(20,23,27)(22,26,24)Polycyclic presentation:
2,3^7|c^3=e,d^3=f,g^3=h,c^b=ce,d^b=df,g^a=g^2h^2,g^b=gh,h^a=h^2234 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/9)×#6900, 3−1+2×#92265, #72243293, … |
| maximal normal subgroup of | AGL(1,ℤ/9)×#6900, 3−1+2×#92265, #72243293, … |
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, 3, F.5);;
gap> SetPower(coll, 4, F.6);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);