Finite Group DatabasebetaFinite Group Databasebeta| order | 20169 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6856 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C83 · Schur multiplier (H2) = C3 |
| direct product | C83×#6859 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C83 | H(3,𝔽3) |
| commutator, Frattini, cosocle | C33 | C32×C83 |
| prime radical | C3×H(3,𝔽3)×C83 | C3 |
| LCS | G ▷ C33 ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C83 ◁ C33×C83 ◁ G | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 6723 |
A minimal faithful permutation representation — 2 generators on 119 points:
(2,4,6)(10,11,14)(12,18,22)(13,17,23)(15,24,20)(16,27,21)(19,29,32)(25,35,36)(26,34,31)(28,33,30)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(1,2,5)(3,6,9)(4,8,7)(10,12,19,21,31,36,13,20,30)(11,15,25,16,18,28,17,26,29)(14,22,32,27,34,35,23,24,33)2905 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20169, 6);