Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C25 |
| direct product | C89×#11176 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | #124 |
| commutator | C22×C4 | C22×C4×C89 |
| Frattini, cosocle | C23×C4 | C23×C89 |
| socle | C22×C89 | #394 |
| prime radical | C22×#41×C89 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ C23×C4×C89 ◁ G | |
834 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 3 generators on 125 points:
(1,2)(3,4)(5,6)(7,17,20,12)(8,21,24,32)(9,15)(10,16)(11,14)(13,23,30,26)(18,29)(19,31,25,28)(22,33)(27,34)(35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125);(5,7)(6,12)(8,22)(9,19)(10,25)(11,20)(13,29)(14,17)(15,28)(16,31)(18,23)(21,27)(24,34)(26,35)(30,36)(32,33);(2,4)(5,8)(6,13,14,30)(7,18)(9,23)(10,26)(11,24)(12,27,17,33)(15,21,16,32)(19,34)(20,35)(22,25)(28,36,31,29)4628 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 4164);