Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C89 · Schur multiplier (H2) = C26 |
| direct product | C89×#33388 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C89 | C22×#102 |
| commutator | C8 | C23×C4×C89 |
| Frattini, cosocle | C2×C8 | C24×C89 |
| prime radical | C89×#3584 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C2×#41×C89 ◁ C2×#131×C89 ◁ G | |
1128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 4 generators on 121 points:
(2,7,9,21)(4,14)(5,17)(8,20,23,31)(10,15,18,22)(12,27)(13,26)(16,19)(24,28,29,32)(25,30)(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)(5,18)(6,9)(7,19)(10,17)(11,23)(12,28)(13,29)(14,21)(16,22)(20,30)(24,26)(25,32)(27,31);(1,3)(2,8)(4,12)(5,13)(6,11)(7,20)(9,23)(10,24)(14,27)(15,28)(16,25)(17,26)(18,29)(19,30)(21,31)(22,32);(3,11)(8,23)(12,25)(13,26)(20,31)(24,29)(27,30)(28,32)5162 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26376);