Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C33 |
No aliases for this group.
| Fitting, Sylow | C32×#6900 ⋊ C2 |
3970 conjugacy classes of subgroups; only the 269 normal subgroups are shown.
This group has 3970 conjugacy classes of subgroups — too many to draw. Only its 269 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 162 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6)(7,13,14,12,10,11,9,8,15)(16,17)(19,24)(20,22)(21,23);(4,5,6)(8,10,13)(11,15,14)(19,20,21)(22,23,24);(1,2,3);(16,21,22,18,20,23,17,19,24)Polycyclic presentation:
2,3^7|c^3=d^2,e^3=h,c^a=ch^2,c^b=cd,e^a=e^2h^2,e^b=eh^2,f^a=f^2,g^a=g^2,h^a=h^2246 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×#92265, #72243916, #72254059, … |
| maximal normal subgroup of | C3⋊D3×#92265, #72243916, #72254059, … |
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.4^2);;
gap> SetPower(coll, 5, F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.8^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.8^2);;
gap> SetConjugate(coll, 5, 2, F.5*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);