Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C89 · Schur multiplier (H2) = C25 |
| direct product | C89×#33404 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C89 | C22×#106 |
| commutator | C8 | C23×C4×C89 |
| Frattini, cosocle | C2×C8 | C24×C89 |
| prime radical | C2×#41×C89 | D4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C2×#41×C89 ◁ C2×#131×C89 ◁ G | |
636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 4 generators on 121 points:
(2,7,9,18)(4,14)(5,16)(8,17,20,28)(10,23,22,19)(12,26)(13,25)(15,27)(21,31,30,29)(24,32)(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);(1,2,4,7,5,10,14,19,6,9,15,18,16,22,27,23)(3,8,12,17,13,21,26,29,11,20,24,28,25,30,32,31);(1,3)(2,8)(4,12)(5,13)(6,11)(7,17)(9,20)(10,21)(14,26)(15,24)(16,25)(18,28)(19,29)(22,30)(23,31)(27,32);(3,11)(8,20)(12,24)(13,25)(17,28)(21,30)(26,32)(29,31)5162 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26392);