Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C23 |
| direct product | C89×#12949 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C89 | #1864 |
| commutator | C2×C8 | C22×C4×C89 |
| Frattini, cosocle | C22×C8 | C23×C89 |
| prime radical | C2×#135×C89 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C22×C4×C89 ◁ #405×C89 ◁ G | |
394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 3 generators on 121 points:
(1,2,6,13,8,14,7,25)(3,15,17,29,19,26,18,30)(4,20,23,12,5,9,21,11)(10,22,27,16,28,24,31,32)(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);(2,9,14,20)(4,21)(5,23)(6,7)(10,26,28,15)(11,13,12,25)(16,22)(17,18)(24,32)(27,29,31,30);(1,3)(2,10)(4,22)(5,24)(6,18)(7,17)(8,19)(9,26)(11,29)(12,30)(13,31)(14,28)(15,20)(16,21)(23,32)(25,27)3560 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 5937);