Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C23 |
| direct product | C89×#32871 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C89 | #2826 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| prime radical | C2×AGL(1,ℤ/8)×C89 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C22×C4×C89 ◁ G | |
492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 4 generators on 121 points:
(1,2,7,9,6,12,5,11)(3,13,14,21,17,23,16,8)(4,10,18,24,20,27,19,26)(15,28,29,25,30,32,31,22)(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);(1,3,5,16,6,17,7,14)(2,8,9,13,12,21,11,23)(4,15,18,29,20,30,19,31)(10,25,26,28,27,22,24,32);(2,9)(3,14)(4,18)(5,7)(8,21)(11,12)(15,30)(16,17)(19,20)(22,28)(24,26)(25,32);(1,4)(2,10)(3,15)(5,19)(6,20)(7,18)(8,22)(9,24)(11,26)(12,27)(13,28)(14,29)(16,31)(17,30)(21,25)(23,32)2492 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 25859);