Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C89 · Schur multiplier (H2) = C8 |
| direct product | C89×#8270 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C89 | C2×D4 |
| commutator | C2×C4 | C2×C42×C89 |
| Frattini, cosocle | C2×C42 | C23×C89 |
| socle | C22×C89 | C4×C4⋊C4 |
| prime radical | C23×C89 | #117 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C89 ◁ C2×C4×C8×C89 ◁ G | |
342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 3 generators on 121 points:
(2,6,8,11)(7,13,14,15)(17,18)(19,22)(20,24)(21,23)(25,29)(26,28)(27,30)(31,32)(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);(1,2)(3,7)(4,11)(5,8)(6,12)(9,15)(10,14)(13,16)(17,19,20,25,21,26,27,31)(18,22,24,29,23,28,30,32);(1,3,4,9,5,10,12,16)(2,7,11,15,8,14,6,13)(18,23)(22,28)(24,30)(29,32)6764 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 1258);