Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#11280 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C89 | #393 |
| commutator | C24 | C22×C4×C89 |
| Frattini, cosocle | C23×C4 | C23×C89 |
| socle | C2×C89 | #1442 |
| prime radical | C24×C89 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C89 ◁ C23×C4×C89 ◁ G | |
580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 3 generators on 121 points:
(1,2,5,10,6,9,20,26)(3,11,14,27,13,7,19,22)(4,15,18,29,17,8,23,25)(12,21,28,31,16,24,30,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(1,3)(2,7)(4,16,17,12)(5,19)(6,13)(8,24,15,21)(9,11)(10,27)(14,20)(18,28,23,30)(22,26)(25,31,29,32);(1,4)(2,8,9,15)(3,12)(5,18)(6,17)(7,21,11,24)(10,25,26,29)(13,16)(14,28)(19,30)(20,23)(22,31,27,32)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 4268);