Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C34 |
No aliases for this group.
| Fitting, Sylow | #14554867 ⋊ C2 |
846 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 846 conjugacy classes of subgroups — too many to draw. Only its 104 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:
(1,4,15)(2,9,27)(3,13,34)(5,41,38,19,17,62)(6,18,37)(7,16,39)(8,23,47)(10,54,52,30,29,72)(11,28,53)(12,33,58)(14,60,59,36,35,75)(20,66,63,44,42,77)(21,65,64,45,40,78)(22,67,61,46,43,76)(24,69,68,49,48,79)(25,26,50)(31,71,70,55,51,80)(32,74,73,56,57,81);(1,20,56)(2,32,19)(3,31,46)(4,35,71)(5,30,25)(6,10,55)(7,14,49)(8,22,36)(9,48,67)(11,24,45)(12,21,44)(13,57,65)(15,63,81)(16,29,74)(17,66,23)(18,42,69)(26,40,60)(27,73,62)(28,51,41)(33,43,54)(34,70,76)(37,52,80)(38,72,50)(39,59,79)(47,61,75)(53,68,78)(58,64,77);(1,3,25,6,11,8,7,2,12)(4,13,26,18,28,23,16,9,33)(5,10,24,22,14,32,21,20,31)(15,34,50,37,53,47,39,27,58)(17,29,48,43,35,57,40,42,51)(19,30,49,46,36,56,45,44,55)(38,52,68,61,59,73,64,63,70)(41,54,69,67,60,74,65,66,71)(62,72,79,76,75,81,78,77,80)Polycyclic presentation:
2,3^7|b^3=h^2,c^3=h^2,c^b=ch^2,d^c=de,e^d=eh^2,f^a=f^2,f^b=fg,f^c=fg^2,g^a=g^2h,g^f=gh114 conjugacy classes — view the full character table.
| maximal subgroup of | #72243655, #72243743 |
| maximal normal subgroup of | #72243655, #72243743 |
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, 2, F.8^2);;
gap> SetPower(coll, 3, F.8^2);;
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.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 3, 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);