Finite Group DatabasebetaFinite Group Databasebeta| order | 21627 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C89 · Schur multiplier (H2) = C33 |
| direct product | C89×#6912 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C3×C89 | C3×H(3,𝔽3) |
| commutator, Frattini, cosocle | C32 | C33×C89 |
| prime radical | C33×C89 | C32 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C89 ◁ C33×C89 ◁ G | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 116 |
| transitive degree | μt(G) = 2403 |
A minimal faithful permutation representation — 3 generators on 116 points:
(1,2,5,4,8,15,13,7,19)(3,9,14,12,6,16,11,18,25)(10,21,24,22,20,26,23,17,27)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116);(1,3,10,13,11,23,4,12,22)(2,6,17,7,9,20,8,18,21)(5,14,24,19,25,27,15,16,26);(3,11,12)(6,9,18)(10,22,23)(14,25,16)(17,21,20)(24,26,27)3115 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21627, 59);