Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C33 |
No aliases for this group.
| Fitting, Sylow | #14552206 ⋊ C2 |
1459 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1459 conjugacy classes of subgroups — too many to draw. Only its 130 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)(3,7)(5,8)(6,9)(10,13,18,12,16,14,15,11,17)(19,20)(21,27)(23,24)(25,26);(1,2,8)(3,5,4)(6,9,7)(11,13,16)(14,18,17)(19,25,24,22,23,26,20,21,27);(2,5,6)(4,8,9)(21,25,23)(24,27,26)Polycyclic presentation:
2,3^7|d^3=e,f^3=h,d^a=dh^2,d^c=de,f^a=f^2h^2,f^b=fg,f^c=fg^2h,g^a=g^2,h^a=h^2159 conjugacy classes — view the full character table.
| maximal subgroup of | #72243278, #72243361, #72243916 |
| maximal normal subgroup of | #72243361 |
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> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2*F.8^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, F.6*F.7^2*F.8);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);