Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C34 |
No aliases for this group.
| Fitting, Sylow | #14554871 ⋊ C2 |
883 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 883 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:
(1,4,16)(2,9,27)(3,13,34)(5,43,40,20,18,64)(6,19,39)(7,17,41)(8,23,47)(10,53,52,29,28,72)(11,15,37)(12,26,50)(14,60,59,36,35,75)(21,66,65,45,42,78)(22,67,63,46,44,77)(24,69,68,49,48,79)(25,33,58)(30,71,70,54,51,80)(31,74,73,55,56,81)(32,62,61,57,38,76);(1,32,55,6,14,54,7,10,49)(2,30,45,11,24,20,3,31,46)(4,28,71,19,38,69,17,35,74)(5,36,12,22,29,8,21,57,25)(9,56,43,15,51,67,13,48,66)(16,61,81,39,59,80,41,52,79)(18,62,23,44,60,33,42,53,26)(27,70,78,37,68,64,34,73,77)(40,75,50,63,72,47,65,76,58);(1,3,25,7,11,8,6,2,12)(4,13,33,17,15,23,19,9,26)(5,32,30,21,10,31,22,14,24)(16,34,58,41,37,47,39,27,50)(18,38,51,42,28,56,44,35,48)(20,57,54,45,29,55,46,36,49)(40,61,70,65,52,73,63,59,68)(43,62,71,66,53,74,67,60,69)(64,76,80,78,72,81,77,75,79)Polycyclic presentation:
2,3^7|b^3=h,c^3=h,c^b=ch,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 | #72243741, #72243838 |
| maximal normal subgroup of | #72243741, #72243838 |
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);;
gap> SetPower(coll, 3, F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
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);