Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C33 |
No aliases for this group.
| Fitting, Sylow | #14554333 ⋊ C2 |
| derived | C3×#733 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×#733 | C2×C32 |
| Frattini | C33 | C32×C3⋊D3 |
| Fitting | #14554333 | C2 |
| socle | C3 | C3×#92300 |
| LCS | G ▷ C3×#733 | |
| UCS | C1 | |
1513 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 1513 conjugacy classes of subgroups — too many to draw. Only its 77 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:
(2,7)(3,37,41,13,15,43)(4,49,21,19,20,50)(5,6)(8,22)(9,12)(10,58,61,28,29,76)(11,53,34,32,25,57)(14,68,44,36,40,39)(16,74,73,46,42,81)(17,45,69,18,48,72)(23,77,78,47,52,63)(24,56,51,33,27,54)(26,65,79,31,55,71)(30,70,66,38,62,75)(35,80,64,60,67,59);(1,14,17,5,18,36,6,3,13)(2,10,38,12,35,47,8,16,31)(4,40,43,21,45,69,20,15,39)(7,26,28,22,23,46,9,30,60)(11,29,71,34,67,75,25,42,63)(19,72,48,49,41,68,50,44,37)(24,55,59,51,52,76,27,62,81)(32,78,58,53,66,74,57,79,80)(33,64,70,56,73,77,54,61,65);(1,8,7,6,12,9,5,2,22)(3,35,30,18,10,23,14,16,26)(4,25,24,20,34,27,21,11,51)(13,31,46,36,47,28,17,38,60)(15,67,62,45,29,52,40,42,55)(19,54,32,50,56,57,49,33,53)(37,65,74,68,77,58,48,70,80)(39,63,76,69,75,59,43,71,81)(41,64,66,72,61,78,44,73,79)Polycyclic presentation:
2,3^7|d^3=h,e^3=h^2,d^a=d^2h^2,d^b=dh,e^a=e^2h,e^b=eh^2,e^c=ef,e^d=eg,f^a=f^2,f^c=fh,g^a=gh^2,g^e=gh,h^a=h^2118 conjugacy classes — view the full character table.
| maximal subgroup of | #72243761, #72243882 |
| maximal normal subgroup of | #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, 4, F.8);;
gap> SetPower(coll, 5, F.8^2);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.8^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5^2*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.8^2);;
gap> SetConjugate(coll, 5, 3, F.5*F.6);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7*F.8^2);;
gap> SetConjugate(coll, 7, 5, F.7*F.8);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);