Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C89 · Schur multiplier (H2) = C26 |
| direct product | C2×C89×#2138 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C89 | D8 |
| commutator | C8 | C23×C4×C89 |
| Frattini, cosocle | C2×C8 | C24×C89 |
| socle | C23×C89 | C2×D8 |
| prime radical | C22×#135×C89 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C89 ◁ C23×C4×C89 ◁ C23×C8×C89 ◁ G | |
1190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 4 generators on 127 points:
(1,2)(39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(3,4,5,6)(7,8)(9,18)(10,15)(11,23)(12,27)(13,17)(14,20)(16,25)(19,34)(21,35)(22,33)(24,29)(26,37)(28,32)(30,38)(31,36);(7,9)(8,14)(10,19)(11,24)(12,28)(13,21)(15,29)(16,32)(17,31)(18,23)(20,25)(22,36)(26,38)(27,30)(33,34)(35,37);(7,10,13,12)(8,15,17,27)(9,19,21,28)(11,22,26,25)(14,29,31,30)(16,23,33,37)(18,34,35,32)(20,24,36,38)7832 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26364);