Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C89 · Schur multiplier (H2) = C23 |
| direct product | C89×#22370 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C89 | C24 |
| commutator | C23 | C23×C4×C89 |
| Frattini, cosocle | C22×C4 | C24×C89 |
| socle | C23×C89 | C23×C4 |
| prime radical | C22×D4×C89 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C89 ◁ G | |
690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 4 generators on 121 points:
(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,18,22,19,21,24,30,26)(20,27,29,32,28,23,31,25)(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);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,24,21,26,30,18)(20,25,29,27,28,32,31,23);(2,5)(6,8)(10,14)(11,16)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,32)(30,31);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(20,28)(23,27)(25,32)(29,31)4628 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 15358);