Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#29385 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C89 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| socle | C2×C89 | #2598 |
| prime radical | C89×#2010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C89 ◁ C22×C4×C89 ◁ G | |
662 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)(3,12)(4,8)(5,9)(6,11)(7,14)(10,18)(13,27)(15,20)(16,24)(17,25)(19,32)(21,30)(22,29)(23,31)(26,28)(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);(1,3,5,14,6,15,18,30)(2,7,10,12,11,21,9,20)(4,13,17,29,16,28,31,32)(8,22,23,27,24,19,25,26);(1,4)(2,8)(3,13)(5,17)(6,16)(7,19)(9,25)(10,23)(11,24)(12,26)(14,29)(15,28)(18,31)(20,27)(21,22)(30,32);(2,9,11,10)(4,16)(7,20,21,12)(8,23,24,25)(13,28)(17,31)(19,26,22,27)(29,32)3560 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 22373);