Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C89 · Schur multiplier (H2) = C25 |
| direct product | C2×C89×#2171 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C89 | C2×D4 |
| commutator | C4 | C23×C8×C89 |
| Frattini, cosocle | C2×C8 | C24×C89 |
| socle | C22×C89 | C8×D4 |
| prime radical | C2×#135×C89 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C89 ◁ C22×C16×C89 ◁ G | |
654 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);(3,4,8,14,9,13,26,32,10,15,27,17,7,23,25,33)(5,16,19,21,18,11,28,22,12,29,31,20,24,30,34,6);(3,5)(4,11)(6,17)(7,24)(8,19)(9,18)(10,12)(13,29)(14,22)(15,30)(16,23)(20,32)(21,33)(25,34)(26,28)(27,31);(3,6,8,16,9,21,26,11,10,22,27,29,7,20,25,30)(4,12,14,31,13,24,32,34,15,5,17,19,23,18,33,28)7832 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26428);