Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C34 |
No aliases for this group.
| Fitting, Sylow | #14554333 ⋊ C2 |
1366 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 1366 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:
(1,12,36,6,16,14,5,3,33)(2,13,47,11,22,28,7,9,35)(4,65,39,42,15,67,20,41,37,17,40,64,19,38,44,45,18,63)(8,27,62,24,34,50,21,25,54)(10,66,57,59,29,78,23,58,55,30,43,76,32,56,61,48,31,75)(26,77,72,69,51,81,46,73,70,52,60,80,49,71,74,68,53,79);(1,23,69,5,32,52,6,10,68)(2,46,17,7,49,45,11,26,42)(3,29,71,16,43,77,12,31,73)(4,48,24,20,59,8,19,30,21)(9,51,65,22,60,41,13,53,38)(14,61,80,36,57,79,33,55,81)(15,56,25,40,66,34,18,58,27)(28,74,63,47,72,67,35,70,64)(37,78,54,44,76,50,39,75,62);(2,7,11)(3,12,16)(8,24,21)(10,23,32)(14,36,33)(15,18,40)(25,34,27)(26,49,46)(28,35,47)(30,48,59)(37,44,39)(38,41,65)(51,60,53)(52,69,68)(55,57,61)(63,67,64)(71,77,73)(75,76,78)Polycyclic presentation:
2,3^7|c^3=h^2,d^3=h,c^b=ch^2,d^b=dh,d^c=de,e^d=eh^2,f^a=f^2,f^d=fg,g^a=g^2h,g^f=gh130 conjugacy classes — view the full character table.
| maximal subgroup of | #72243741, #72243882 |
| maximal normal subgroup of | #72243741, #72243882 |
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> SetPower(coll, 4, F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8^2);;
gap> SetConjugate(coll, 4, 2, F.4*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, 4, 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);