Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C89 · Schur multiplier (H2) = C25 |
| direct product | C89×#22690 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | D42 |
| commutator | C2×C4 | C23×C4×C89 |
| Frattini, cosocle | C22×C4 | C24×C89 |
| prime radical | C2×#135×C89 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C22×C4×C89 ◁ G | |
846 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)(7,19)(8,23)(9,15)(10,17)(11,16)(12,18)(13,21)(14,24)(20,34)(22,32)(25,33)(26,36)(27,30)(28,31)(29,35)(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);(5,7,12,22)(6,13,18,31)(8,9,25,27)(10,28,11,21)(14,30,34,15)(16,32,17,19)(20,29,24,26)(23,36,33,35);(5,8,11,24,12,25,10,20)(6,14,17,23,18,34,16,33)(7,9,21,26,22,27,28,29)(13,30,19,36,31,15,32,35);(5,9)(6,15)(7,20)(8,21)(10,29)(11,26)(12,27)(13,23)(14,32)(16,36)(17,35)(18,30)(19,34)(22,24)(25,28)(31,33)5162 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 15678);