Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C28 |
| direct product | C89×#60986 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C89 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×#117×C89 ◁ G | |
1746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 5 generators on 121 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,28)(21,26)(23,25)(24,29)(27,31)(30,32)(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);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,22,18)(20,24,30,27)(21,26,23,25)(28,31,32,29);(3,10)(6,9)(11,16)(13,15)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32);(2,7)(6,9)(8,14)(13,15)(17,21,22,23)(18,25,19,26)(20,29,30,31)(24,28,27,32);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(17,21,22,23)(18,26,19,25)(20,29,30,31)(24,32,27,28)4361 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53974);