Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C25 |
| direct product | C89×#11178 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C89 | #388 |
| commutator | C22×C4 | C22×C4×C89 |
| Frattini, cosocle | C23×C4 | C23×C89 |
| socle | C2×C89 | #1440 |
| prime radical | C22×#41×C89 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C89 ◁ C23×C4×C89 ◁ G | |
820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 3 generators on 121 points:
(1,2)(3,13)(4,18,22,26)(5,11)(6,10)(7,12)(8,15)(9,20,27,21)(14,29,31,23)(16,24)(17,28)(19,32,30,25)(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)(2,8)(4,19)(5,16)(6,15)(7,17)(9,25)(10,28)(11,13)(12,24)(14,21)(18,29)(20,31)(22,30)(23,26)(27,32);(1,4,5,20,7,22,6,21)(2,9,10,26,12,27,11,18)(3,14,15,30,17,31,16,19)(8,23,13,25,24,29,28,32)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 4166);