Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C89 · Schur multiplier (H2) = C23 |
| direct product | C89×#12792 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C89 | #121 |
| commutator | C2×C4 | C22×C8×C89 |
| Frattini, cosocle | C22×C8 | C23×C89 |
| socle | C2×C89 | #1797 |
| prime radical | C2×#41×C89 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C89 ◁ C22×C8×C89 ◁ G | |
440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 3 generators on 121 points:
(1,2,6,13,7,11,18,26,8,14,17,9,5,12,3,15)(4,19,22,31,21,28,16,30,23,29,24,32,20,10,27,25)(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);(1,3,5,17,8,18,7,6)(2,9,11,15,14,13,12,26)(4,16,20,22,23,27,21,24)(10,25,19,31,28,30,29,32);(1,4)(2,10)(3,16)(5,20)(6,24)(7,21)(8,23)(9,25)(11,19)(12,29)(13,30)(14,28)(15,31)(17,22)(18,27)(26,32)5696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 5780);