Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C34 |
No aliases for this group.
| Fitting, Sylow | #14554345 ⋊ C2 |
1384 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 1384 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 81 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,3,14)(2,9,28)(4,43,45,17,19,67)(5,13,35)(6,16,37)(7,22,48)(8,25,36)(10,60,62,30,32,78)(11,12,34)(15,64,18,39,40,42)(20,65,38,46,41,63)(21,27,54)(23,72,74,50,51,81)(24,33,55)(26,57,58,53,29,76)(31,66,56,59,44,75)(47,77,69,68,61,79)(49,80,52,70,71,73);(1,4,42,5,20,17,6,18,46)(2,26,30,11,10,59,8,31,53)(3,19,43,13,41,65,16,15,39)(7,47,68,21,52,73,24,23,50)(9,29,66,12,32,57,25,44,60)(14,38,67,35,40,63,37,45,64)(22,61,70,27,49,72,33,51,77)(28,62,75,34,56,76,36,58,78)(48,71,80,54,74,81,55,69,79);(1,2,7)(3,12,33)(4,10,23)(5,11,21)(6,8,24)(9,27,16)(13,25,22)(14,36,54)(15,32,61)(17,30,50)(18,26,52)(19,44,49)(20,31,47)(28,55,35)(29,51,41)(34,48,37)(38,62,71)(39,60,77)(40,56,74)(42,53,73)(43,66,70)(45,58,69)(46,59,68)(57,72,65)(63,78,80)(64,75,81)(67,76,79)Polycyclic presentation:
2,3^7|c^b=ch^2,d^b=dh^2,d^c=de,e^d=eh,f^a=f^2,f^c=fg,f^d=fg^2,g^a=g^2h,g^f=gh130 conjugacy classes — view the full character table.
| maximal subgroup of | #72243652, #72243837 |
| maximal normal subgroup of | #72243652, #72243837 |
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, 3, 2, F.3*F.8^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.8^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.7^2);;
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);