Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C25 |
| direct product | C89×#33836 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C89 |
| prime radical | C89×#3285 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C4×D4×C89 ◁ C8×D4×C89 ◁ G | |
664 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 4 generators on 125 points:
(1,2)(3,4)(6,11)(7,15)(8,19)(9,22)(12,29)(13,33)(14,28)(16,32)(17,30)(18,25)(20,24)(23,26)(27,31)(34,35)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(5,6,10,14)(7,16,18,29)(8,20,23,33)(9,24,22,13)(11,26,28,19)(12,30,32,17)(15,35,25,27)(21,34,36,31);(2,4);(5,7,10,18)(6,12,14,32)(8,21,23,36)(9,25,22,15)(11,27,28,35)(13,34,24,31)(16,33,29,20)(17,26,30,19)3827 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26824);