Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C26 |
| direct product | C2×C89×#2049 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | #394 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| prime radical | C22×#41×C89 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C23×C4×C89 ◁ G | |
1322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 4 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);(3,4,9,14)(5,15,18,30)(6,19,23,28)(7,13,8,12)(10,24,26,17)(11,21,29,22)(16,27,20,25)(31,34,33,32);(3,5)(4,10)(6,20)(7,24)(8,17)(9,18)(11,27)(12,30)(13,15)(14,26)(16,23)(19,33)(21,32)(22,34)(25,29)(28,31);(3,6,7,21,9,23,8,22)(4,11,12,28,14,29,13,19)(5,16,17,32,18,20,24,34)(10,25,15,31,26,27,30,33)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 16512);