Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C34 |
No aliases for this group.
| Fitting, Sylow | #14552223 ⋊ C2 |
1369 conjugacy classes of subgroups; only the 128 normal subgroups are shown.
This group has 1369 conjugacy classes of subgroups — too many to draw. Only its 128 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:
(2,7,9,4,5,8)(3,6)(11,14,15)(13,17,18)(19,25,23,22,20,24,26,21,27);(1,2,8)(3,5,7)(4,6,9)(10,11,13)(12,15,17)(14,18,16);(2,5,9)(4,7,8)(19,21,27,22,25,23,26,20,24)Polycyclic presentation:
2,3^7|b^3=e,d^3=e,d^b=de,d^c=df,g^a=g^2,g^b=gh,g^c=gh,g^d=gh,h^a=h^2186 conjugacy classes — view the full character table.
| maximal subgroup of | #72243516 |
| maximal normal subgroup of | #72243516 |
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.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 7, 3, 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);