Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C33 |
No aliases for this group.
| Fitting, Sylow | #14554376 ⋊ C2 |
1630 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 1630 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 54 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,6,5)(4,9,8)(11,22,28,14,16,24)(12,17)(13,20)(15,33,30,21,26,34)(18,29)(19,31)(23,36,32,27,35,25);(2,5,6)(4,8,9)(10,11,24)(12,15,32)(13,16,22)(14,20,28)(17,27,34)(18,30,23)(19,26,35)(21,31,25)(29,36,33);(1,6,9,3,5,4,7,2,8)(11,15,25)(12,18,19)(14,21,32)(16,26,27)(17,29,31)(22,33,23)(24,34,35)(28,30,36)Polycyclic presentation:
2,3^7|b^3=e,c^b=ce,d^b=de^2,f^a=f^2,f^b=fg,f^c=fh,f^d=fh,g^a=g^2,g^b=gh^2,h^a=h^2135 conjugacy classes — view the full character table.
| maximal subgroup of | #72277763 |
| maximal normal subgroup of | #72277763 |
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> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.7*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);