Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C29 |
| direct product | C2×C89×#3416 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C25×C89 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×#41×C89 ◁ C2×#131×C89 ◁ G | |
2074 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 5 generators on 123 points:
(1,2)(35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(3,4,8,13)(5,10,14,24)(6,9,19,23)(7,20,18,12)(11,27,22,17)(15,21,28,32)(16,31,29,26)(25,34,33,30);(4,9)(6,17)(7,18)(10,21)(11,12)(13,23)(15,30)(16,29)(19,27)(20,22)(24,32)(25,26)(28,34)(31,33);(3,5)(4,10)(6,15)(7,16)(8,14)(9,21)(11,25)(12,26)(13,24)(17,30)(18,29)(19,28)(20,31)(22,33)(23,32)(27,34);(5,14)(10,24)(15,28)(16,29)(21,32)(25,33)(26,31)(30,34)5696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 55621);