Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C34 |
No aliases for this group.
| Fitting, Sylow | #14555438 ⋊ C2 |
4072 conjugacy classes of subgroups; only the 219 normal subgroups are shown.
This group has 4072 conjugacy classes of subgroups — too many to draw. Only its 219 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 — 4 generators on 81 points:
(1,7,22,6,11,8,5,2,21)(3,51,26,35,9,58,19,32,23,12,30,49,15,25,36,34,13,46)(4,57,29,43,10,69,20,42,24,16,31,56,18,28,44,45,17,48)(14,63,62,66,55,71,41,53,52,38,37,80,40,77,47,65,27,75)(33,78,73,67,50,81,60,64,70,39,74,76,61,54,79,68,59,72);(2,7,11)(3,15,19)(8,22,21)(9,30,13)(10,17,31)(12,34,35)(14,40,41)(24,44,29)(27,55,37)(28,42,57)(33,60,61)(38,65,66)(39,67,68)(46,49,58)(48,69,56)(54,78,64)(70,73,79)(71,75,80);(1,4,16)(2,10,28)(3,14,39)(5,20,45)(6,18,43)(7,17,42)(8,24,48)(9,27,54)(11,31,57)(12,33,38)(13,37,64)(15,40,67)(19,41,68)(21,29,56)(22,44,69)(23,47,72)(25,50,53)(26,52,76)(30,55,78)(32,59,63)(34,60,65)(35,61,66)(36,62,81)(46,70,71)(49,73,75)(51,74,77)(58,79,80);(1,3,12)(2,9,25)(4,14,33)(5,19,34)(6,15,35)(7,13,32)(8,23,46)(10,27,50)(11,30,51)(16,39,38)(17,37,59)(18,40,61)(20,41,60)(21,26,49)(22,36,58)(24,47,70)(28,54,53)(29,52,73)(31,55,74)(42,64,63)(43,67,66)(44,62,79)(45,68,65)(48,72,71)(56,76,75)(57,78,77)(69,81,80)Polycyclic presentation:
2,3^7|c^3=h^2,c^b=ch^2,d^a=d^2,d^c=dg,e^a=e^2,e^c=eg^2h^2,f^a=f^2,f^d=fh,g^a=g^2h^2,g^d=gh^2,g^e=gh166 conjugacy classes — view the full character table.
| maximal subgroup of | #72243644, #72243787, #72243831 |
| maximal normal subgroup of | #72243644, #72243787, #72243831 |
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.8^2);;
gap> SetConjugate(coll, 3, 2, F.3*F.8^2);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.7^2*F.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 7, 4, F.7*F.8^2);;
gap> SetConjugate(coll, 7, 5, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);