Finite Group DatabasebetaFinite Group Databasebeta| order | 21627 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C9×C89 · Schur multiplier (H2) = C32 |
| direct product | C89×#6868 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C89 | H(3,𝔽3) |
| commutator | C32 | C3×C9×C89 |
| Frattini, cosocle | C33 | C32×C89 |
| prime radical | C33×C89 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C89 ◁ C33×C89 ◁ G | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 7209 |
A minimal faithful permutation representation — 2 generators on 125 points:
(1,2,4,3,5,7,6,8,9)(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)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(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)4539 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21627, 15);