Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C23×C4 |
| direct product | C89×#29154 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C89 | #555 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| socle | C2×C89 | #2563 |
| prime radical | C2×#135×C89 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C89 ◁ C22×C4×C89 ◁ G | |
612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 4 generators on 121 points:
(1,2,6,12)(3,9)(4,16,18,22)(5,8,7,11)(10,19,26,17)(13,21)(14,28)(15,23)(20,25)(24,27)(29,31)(30,32)(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);(2,8,12,11)(3,13,15,20)(9,23)(10,22,26,16)(14,27,29,30)(21,25)(24,32)(28,31);(1,3)(2,9)(4,14)(5,13)(6,15)(7,20)(8,21)(10,24)(11,25)(12,23)(16,28)(17,27)(18,29)(19,30)(22,31)(26,32);(1,4,5,17,6,18,7,19)(2,10,11,16,12,26,8,22)(3,14,13,27,15,29,20,30)(9,24,25,28,23,32,21,31)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 22142);