Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C89 · Schur multiplier (H2) = C24×C4 |
| direct product | C89×#10941 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C89 | #386 |
| commutator | C23 | C2×C42×C89 |
| Frattini, cosocle | C23×C4 | C23×C89 |
| socle | C2×C89 | #1437 |
| prime radical | C22×#41×C89 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C89 ◁ C23×C4×C89 ◁ G | |
904 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 3 generators on 121 points:
(1,2,5,10,6,9,17,23)(3,7,13,20,12,19,16,24)(4,8,15,21,14,22,28,32)(11,25,27,31,26,18,29,30)(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)(2,7,9,19)(4,11)(5,16)(6,12)(8,18,22,25)(10,24,23,20)(13,17)(14,26)(15,29)(21,31,32,30)(27,28);(1,4,5,15,6,14,17,28)(2,8,10,21,9,22,23,32)(3,11,16,29,12,26,13,27)(7,18,24,31,19,25,20,30)5162 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 3929);