Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C34 |
No aliases for this group.
| Fitting, Sylow | #14554289 ⋊ C2 |
2494 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 2494 conjugacy classes of subgroups — too many to draw. Only its 119 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)(12,19)(17,27)(20,29)(21,30)(22,32)(24,33)(26,34)(28,35)(31,36);(1,2,4)(3,6,8)(5,9,7)(10,20,34)(11,28,30)(12,33,25)(13,17,36)(14,24,35)(15,26,19)(16,22,29)(18,31,32)(21,27,23);(11,15,18)(16,25,23)(17,24,20)(26,28,31)(27,33,29)(34,35,36)Polycyclic presentation:
2,3^7|c^3=h,c^b=ch^2,d^c=de,f^a=f^2,f^c=fg,g^a=g^2h,g^f=gh162 conjugacy classes — view the full character table.
| maximal subgroup of | #72277465 |
| maximal normal subgroup of | #72277465 |
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.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);