Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C9 · Schur multiplier (H2) = C35 |
No aliases for this group.
| Fitting, Sylow | #14552075 ⋊ C2 |
1603 conjugacy classes of subgroups; only the 185 normal subgroups are shown.
This group has 1603 conjugacy classes of subgroups — too many to draw. Only its 185 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:
(1,5,9,3,8,4,6,2,7)(11,16,18,13,14,17)(12,15);(10,11,13)(12,14,17)(15,18,16)(20,23,24)(22,26,27);(11,14,18)(13,16,17)(19,20,22)(21,24,26)(23,27,25)Polycyclic presentation:
2,3^7|d^3=e,c^b=cf,g^a=g^2,g^b=gh,g^d=gh,h^a=h^2330 conjugacy classes — view the full character table.
| maximal subgroup of | #72243516, #72269264, #72269364 |
| maximal normal subgroup of | #72243516, #72269264, #72269364 |
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, 4, F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);