Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C34 |
No aliases for this group.
| Fitting, Sylow | C32⋊H(5,𝔽3) ⋊ C2 |
6154 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 6154 conjugacy classes of subgroups — too many to draw. Only its 237 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) = 81 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,5,6)(4,8,9)(11,15,17)(12,18)(14,22,24)(16,33,27,26,25,34)(19,29)(20,28)(23,35,32,31,30,36);(1,2,4)(3,6,8)(5,9,7)(12,19,20)(16,25,27)(18,28,29)(23,32,30)(26,34,33)(31,35,36);(10,16,31)(11,23,18)(12,26,14)(13,27,35)(15,32,29)(17,30,28)(19,33,24)(20,34,22)(21,25,36)Polycyclic presentation:
2,3^7|d^b=dh,d^c=de,f^a=f^2,g^a=g^2,g^f=gh234 conjugacy classes — view the full character table.
| maximal subgroup of | #72277745, #72277786, #72277801 |
| maximal normal subgroup of | #72277745, #72277786, #72277801 |
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> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);