Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C89 · Schur multiplier (H2) = C25 |
| direct product | C2×C89×#2178 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C89 | D8 |
| commutator | C8 | C23×C4×C89 |
| Frattini, cosocle | C2×C8 | C24×C89 |
| socle | C22×C89 | C4×D8 |
| prime radical | C2×#508×C89 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C89 ◁ C22×C8×C89 ◁ C2×C4×C8×C89 ◁ G | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 4 generators on 123 points:
(1,2)(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);(3,4,8,12,9,14,24,13)(5,15,17,26,19,32,18,34)(6,11,20,29,21,30,7,22)(10,25,27,33,28,23,31,16);(3,5)(4,10)(6,16)(7,23)(8,17)(9,19)(11,26)(12,27)(13,31)(14,28)(15,22)(18,24)(20,25)(21,33)(29,32)(30,34);(3,6,8,20,9,21,24,7)(4,11,12,29,14,30,13,22)(5,16,17,25,19,33,18,23)(10,26,27,32,28,34,31,15)7832 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26449);