Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C89 · Schur multiplier (H2) = C22×C4 |
| direct product | C89×#22392 |
| 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 | |
732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 4 generators on 121 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,25,20,24,22,29)(19,26,28,32,27,23,30,31)(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);(1,3,6,2)(4,7,5,8)(17,18,22,29,20,24,21,25)(19,23,30,32,27,26,28,31);(1,4,6,5)(2,7,3,8)(10,13)(14,16)(17,19,20,27)(18,23,24,26)(21,28,22,30)(25,31,29,32);(1,5,6,4)(2,7,3,8)(9,11,12,15)(10,14,13,16)(18,24)(19,27)(25,29)(28,30)5696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 15380);