Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C89 · Schur multiplier (H2) = C25×C4 |
| direct product | C89×#20420 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C89 | C2×D4 |
| commutator | C4 | C22×C42×C89 |
| Frattini, cosocle | C22×C4 | C24×C89 |
| socle | C22×C89 | C2×C4×D4 |
| prime radical | C22×#41×C89 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C89 ◁ C23×C8×C89 ◁ G | |
1474 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 4 generators on 125 points:
(1,2)(3,4)(5,6,10,14,11,15,9,24)(7,16,20,26,18,30,19,12)(8,13,22,27,23,28,21,34)(17,29,33,35,31,36,32,25)(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);(2,4)(5,7)(6,12,15,26)(8,17)(9,19)(10,20)(11,18)(13,25,28,35)(14,16,24,30)(21,32)(22,33)(23,31)(27,29,34,36);(5,8,10,22,11,23,9,21)(6,13,14,27,15,28,24,34)(7,17,20,33,18,31,19,32)(12,25,16,29,26,35,30,36);(7,18)(12,26)(16,30)(17,31)(19,20)(25,35)(29,36)(32,33)7832 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 13408);