Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C32 |
No aliases for this group.
| Fitting, Sylow | #14552206 ⋊ C2 |
1420 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 1420 conjugacy classes of subgroups — too many to draw. Only its 149 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:
(2,4)(3,6)(5,7)(8,9)(19,22,27,21,25,23,24,20,26);(1,2,4)(3,5,8)(6,9,7)(11,14,18)(13,16,17)(20,22,25)(23,27,26);(2,5,9)(4,7,8)(10,14,16,12,18,13,15,11,17)(20,22,25)(23,27,26)Polycyclic presentation:
2,3^7|b^3=f,d^3=e,c^b=cf^2,d^b=de,d^c=de,g^a=g^2,g^b=gh,h^a=h^2186 conjugacy classes — view the full character table.
| maximal subgroup of | #72243705, #72254047 |
| maximal normal subgroup of | #72243705, #72254047 |
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, 2, F.6);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);