Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#33428 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C89 | #117 |
| commutator | C2×C4 | C23×C4×C89 |
| Frattini, cosocle | C2×C8 | C24×C89 |
| socle | C2×C89 | C2×#432 |
| prime radical | C2×#41×C89 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C89 ◁ #405×C89 ◁ G | |
528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 4 generators on 121 points:
(1,2,4,10,5,11,17,28,6,9,18,24,19,25,31,32)(3,12,14,26,15,22,30,27,13,7,16,20,29,8,21,23)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121);(1,3)(2,7)(4,16)(5,15)(6,13)(8,11)(9,12)(10,26)(14,18)(17,21)(19,29)(20,24)(22,25)(23,32)(27,28)(30,31);(1,3)(2,8,9,22)(4,16)(5,15)(6,13)(7,11,12,25)(10,27,24,23)(14,18)(17,21)(19,29)(20,32,26,28)(30,31);(3,13)(7,12)(8,22)(14,16)(15,29)(20,26)(21,30)(23,27)5696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26416);