Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C34 |
No aliases for this group.
| Fitting, Sylow | #14554344 ⋊ C2 |
1483 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 1483 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:
(1,3,13)(2,9,25)(4,34,33,16,14,56)(5,12,30)(6,15,32)(7,20,41)(8,23,45)(10,51,48,28,27,68)(11,26,49)(17,57,55,36,35,72)(18,54,53,37,31,71)(19,38,58)(21,64,63,43,42,77)(22,40,61)(24,67,66,47,46,78)(29,70,69,52,50,79)(39,74,73,60,59,80)(44,76,75,65,62,81);(1,4,36,5,18,16,6,17,37)(2,24,28,11,10,52,8,29,47)(3,31,54,12,35,57,15,14,34)(7,39,60,19,44,65,22,21,43)(9,27,67,26,50,51,23,46,70)(13,33,72,30,53,56,32,55,71)(20,62,64,38,42,74,40,59,76)(25,66,68,49,48,79,45,69,78)(41,73,80,58,75,81,61,63,77);(1,2,7)(3,9,20)(4,10,21)(5,11,19)(6,8,22)(12,26,38)(13,25,41)(14,27,42)(15,23,40)(16,28,43)(17,24,44)(18,29,39)(30,49,58)(31,50,59)(32,45,61)(33,48,63)(34,51,64)(35,46,62)(36,47,65)(37,52,60)(53,69,73)(54,70,74)(55,66,75)(56,68,77)(57,67,76)(71,79,80)(72,78,81)Polycyclic presentation:
2,3^7|c^b=ch^2,d^c=de,e^d=eh,f^a=f^2,f^c=fg,g^a=g^2h,g^f=gh130 conjugacy classes — view the full character table.
| maximal subgroup of | #72243652, #72243734, #72243833, … |
| maximal normal subgroup of | #72243652, #72243734, #72243833, … |
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, 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, 7, 1, F.7^2*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);