Finite Group DatabasebetaFinite Group Databasebeta| order | 4374 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C33 |
No aliases for this group.
| Fitting, Sylow | #14554343 ⋊ C2 |
| derived | C3×#732 ⋊ C2×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×#732 | C2×C32 |
| Frattini | C33 | C32×C3⋊D3 |
| Fitting | #14554343 | C2 |
| socle | C3 | C3×#92300 |
| LCS | G ▷ C3×#732 | |
| UCS | C1 | |
1636 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 1636 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:
(2,22,26,7,9,27)(3,13)(4,47,19,17,18,48)(5,6)(8,49,29,21,25,24)(10,30,50,12,31,53)(11,56,54,33,28,78)(14,69,59,40,41,64)(15,72,45,42,43,76)(16,46)(20,39)(23,58,55,35,51,79)(32,81,57,62,61,73)(34,66,52,37,65,77)(36,80,71,63,60,70)(38,75,68,44,67,74);(1,3,13)(2,14,62)(4,15,38)(5,20,46)(6,16,39)(7,34,63)(8,36,35)(9,41,81)(10,23,40)(11,37,12)(17,44,42)(18,43,67)(19,45,68)(21,32,33)(22,65,80)(24,57,78)(25,60,58)(26,59,73)(27,52,70)(28,66,30)(29,71,79)(31,51,69)(47,75,72)(48,74,76)(49,61,56)(50,55,64)(53,54,77);(1,8,7,6,12,10,5,2,21)(3,36,34,16,11,23,20,14,32)(4,25,24,19,30,27,18,9,50)(13,62,40,39,35,33,46,37,63)(15,60,57,45,28,52,43,41,55)(17,53,22,47,26,31,48,29,49)(38,81,70,68,58,64,67,66,78)(42,79,69,72,77,56,76,73,80)(44,54,65,75,59,51,74,71,61)Polycyclic presentation:
2,3^7|d^3=h,d^a=d^2h^2,d^b=dh,d^c=df,e^a=e^2,e^d=eg,f^a=f^2,f^c=fh^2,g^a=gh^2,g^e=gh,h^a=h^2118 conjugacy classes — view the full character table.
| maximal subgroup of | #72243693, #72243832, #72243938, … |
| maximal normal subgroup of | #72243832, #72243938, #72243999, … |
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> SetConjugate(coll, 4, 1, F.4^2*F.8^2);;
gap> SetConjugate(coll, 4, 2, F.4*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
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.8^2);;
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);