Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#12986 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C89 | #386 |
| commutator | C2×C4 | C22×C8×C89 |
| Frattini, cosocle | C22×C8 | C23×C89 |
| socle | C2×C89 | #1869 |
| prime radical | C2×#135×C89 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C89 ◁ C22×C8×C89 ◁ G | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 3 generators on 121 points:
(1,2,7,12,6,11,25,29,8,13,27,31,5,24,15,16)(3,14,18,32,19,4,21,23,20,22,10,30,17,9,28,26)(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);(1,3,15,28,5,17,27,10,8,20,25,21,6,19,7,18)(2,9,29,32,24,22,12,26,13,4,16,30,11,14,31,23);(1,4)(2,10)(3,16)(5,14)(6,22)(7,26)(8,9)(11,28)(12,19)(13,18)(15,30)(17,31)(20,29)(21,24)(23,27)(25,32)5162 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 5974);