Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C89 · Schur multiplier (H2) = C25 |
| direct product | C89×#33463 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C89 | C2×D8 |
| commutator | C8 | C23×C4×C89 |
| Frattini, cosocle | C2×C8 | C24×C89 |
| socle | C2×C89 | C2×C4×D8 |
| prime radical | C89×#3414 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C89 ◁ C22×C8×C89 ◁ C2×C4×C8×C89 ◁ G | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 4 generators on 121 points:
(1,2,6,11,7,10,22,12)(3,13,15,24,17,30,16,32)(4,9,18,28,19,27,5,20)(8,23,26,31,25,21,29,14)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(1,3)(2,8)(4,14)(5,21)(6,15)(7,17)(9,24)(10,25)(11,26)(12,29)(13,20)(16,22)(18,23)(19,31)(27,32)(28,30);(1,4,6,18,7,19,22,5)(2,9,11,28,10,27,12,20)(3,14,15,23,17,31,16,21)(8,24,26,30,25,32,29,13);(2,10)(8,25)(9,27)(11,12)(13,30)(20,28)(24,32)(26,29)5696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26451);