Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C2×C4 |
| direct product | C89×#13139 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | #481 |
| commutator | C2×C8 | C22×C4×C89 |
| Frattini, cosocle | C22×C8 | C23×C89 |
| prime radical | C23×C89 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C22×C4×C89 ◁ C2×#97×C89 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 125 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 3 generators on 125 points:
(2,3)(5,6)(7,15,18,31)(8,11)(9,21)(10,24)(12,17,26,33)(13,23)(14,16)(19,34,25,29)(20,35)(22,28,36,30)(27,32)(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);(1,2,4,3)(6,11)(9,20)(12,26)(14,24)(17,33)(22,36)(23,32)(28,30);(1,3,4,2)(5,7,16,19,8,18,10,25)(6,12,24,28,11,26,14,30)(9,22,32,33,20,36,23,17)(13,29,35,31,27,34,21,15)3560 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 6127);