Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#11273 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C89 | #394 |
| commutator | C22×C4 | C22×C4×C89 |
| Frattini, cosocle | C23×C4 | C23×C89 |
| socle | C2×C89 | #1442 |
| prime radical | C24×C89 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C89 ◁ C23×C4×C89 ◁ G | |
576 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)(3,7,12,22)(4,14)(5,18)(6,9)(8,16)(10,19)(11,25,15,30)(13,29,28,21)(17,31)(20,32,23,27)(24,26)(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);(1,3,5,13,6,12,19,28)(2,7,10,21,9,22,18,29)(4,15,16,32,17,11,26,27)(8,23,14,30,24,20,31,25);(1,4)(2,8,9,24)(3,11,12,15)(5,16)(6,17)(7,20)(10,14,18,31)(13,27,28,32)(19,26)(21,25)(22,23)(29,30)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 4261);