Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#33637 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C89 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C24×C89 |
| socle | C2×C89 | C2×#476 |
| prime radical | C2×#135×C89 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C89 ◁ C22×C4×C89 ◁ C2×C4×C8×C89 ◁ G | |
524 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)(3,15)(4,19,22,32)(5,11)(6,13)(7,23)(8,14)(9,17)(10,16,26,31)(12,18)(20,29,21,27)(24,28,30,25)(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);(1,3,6,17,8,18,5,7)(2,9,11,15,14,23,13,12)(4,16,21,30,22,31,20,24)(10,19,25,29,26,32,28,27);(1,4)(2,10)(3,16)(5,20)(6,21)(7,24)(8,22)(9,19)(11,25)(12,27)(13,28)(14,26)(15,29)(17,30)(18,31)(23,32);(1,5,8,6)(2,11,14,13)(3,7,18,17)(4,20,22,21)(9,15,23,12)(10,25,26,28)(16,24,31,30)(19,29,32,27)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26625);