Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C89 · Schur multiplier (H2) = C23×C4 |
| direct product | C89×#20521 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C89 | C24 |
| commutator | C22 | C22×C42×C89 |
| Frattini, cosocle | C22×C4 | C24×C89 |
| socle, prime radical | C23×C89 | C23×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C89 ◁ G | |
866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 22784 |
A minimal faithful permutation representation — 4 generators on 117 points:
(5,6)(7,11)(8,12)(9,10)(13,14,18,21,17,20,26,16)(15,22,25,24,23,19,27,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(1,2,4,3)(6,10)(11,12)(13,15,17,23)(14,19,20,22)(16,24,21,28)(18,25,26,27);(1,3,4,2)(5,7,9,8)(6,11,10,12)(14,20)(15,23)(16,21)(25,27);(5,8,9,7)(6,12,10,11)(13,16,26,20,17,21,18,14)(15,24,27,22,23,28,25,19)8900 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 13509);