Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C89 · Schur multiplier (H2) = C25 |
| direct product | C89×#33469 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C89 | C2×#442 |
| commutator | C8 | C23×C4×C89 |
| Frattini, cosocle | C2×C8 | C24×C89 |
| prime radical | C89×#3414 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C22×C4×C89 ◁ #405×C89 ◁ G | |
646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 4 generators on 121 points:
(2,7,9,22)(3,11)(4,14)(5,18)(8,24,25,32)(10,15,19,23)(12,29)(13,16)(17,20)(21,28,26,27)(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);(1,2)(3,8)(4,15)(5,19)(6,9)(7,20)(10,18)(11,28)(12,25)(13,24)(14,22)(16,32)(17,23)(21,30)(26,31)(27,29);(1,3,5,11,6,12,18,29)(2,8,19,28,9,25,10,27)(4,16,14,31,17,13,20,30)(7,21,15,32,22,26,23,24);(3,12)(8,25)(11,29)(13,16)(21,26)(24,32)(27,28)(30,31)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26457);