Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#33427 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C89 | C2×#117 |
| commutator | C2×C4 | C23×C4×C89 |
| Frattini, cosocle | C2×C8 | C24×C89 |
| socle | C2×C89 | C2×#432 |
| prime radical | C24×C89 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C89 ◁ #405×C89 ◁ G | |
524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 4 generators on 121 points:
(1,2,4,9,5,10,17,25,6,7,15,20,18,21,30,32)(3,11,13,27,14,19,29,24,12,26,16,31,28,8,22,23)(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);(2,7)(4,15)(8,19)(10,21)(11,26)(13,16)(17,30)(22,29);(1,3)(2,8,7,19)(4,16)(5,14)(6,12)(9,24,20,23)(10,11,21,26)(13,15)(17,22)(18,28)(25,31,32,27)(29,30);(3,12)(8,19)(11,26)(13,16)(14,28)(22,29)(23,24)(27,31)5162 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26415);