Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33 · Schur multiplier (H2) = C34 |
No aliases for this group.
| Fitting, Sylow | #14555436 ⋊ C2 |
3820 conjugacy classes of subgroups; only the 225 normal subgroups are shown.
This group has 3820 conjugacy classes of subgroups — too many to draw. Only its 225 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:
(1,3,11)(2,8,25)(4,34,33,13,12,56)(5,10,30)(6,18,40)(7,22,45)(9,51,50,27,26,70)(14,57,55,37,35,72)(15,54,53,36,31,71)(16,29,32)(17,38,58)(19,64,61,43,42,75)(20,41,62)(21,24,48)(23,67,66,47,46,78)(28,69,68,52,49,79)(39,74,73,60,59,80)(44,77,76,65,63,81);(2,7,21)(3,10,29)(4,14,15)(6,17,20)(8,24,22)(13,36,37)(19,44,39)(25,45,48)(26,46,49)(27,52,47)(33,55,53)(34,57,54)(40,58,62)(42,59,63)(56,71,72)(61,76,73)(64,77,74)(70,79,78);(1,9,43)(2,19,13)(3,26,64)(4,27,6)(5,23,65)(7,44,36)(8,42,34)(10,46,77)(11,50,75)(12,51,18)(14,52,17)(15,47,20)(16,28,60)(21,39,37)(22,63,54)(24,59,57)(25,61,56)(29,49,74)(30,66,81)(31,67,41)(32,68,80)(33,70,40)(35,69,38)(45,76,71)(48,73,72)(53,78,62)(55,79,58)Polycyclic presentation:
2,3^7|c^b=ch^2,d^c=de,e^d=eh^2,f^a=f^2,g^a=g^2,g^f=gh202 conjugacy classes — view the full character table.
| maximal subgroup of | #72243652, #72243728, #72243843, … |
| maximal normal subgroup of | #72243652, #72243728, #72243843, … |
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.8^2);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);