Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C25 |
| direct product | C89×#12780 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C89 | #1790 |
| commutator | C2×Q8 | C22×C4×C89 |
| Frattini, cosocle | C2×#41 | C23×C89 |
| prime radical | C89×#3551 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C2×C4×C89 ◁ C2×#41×C89 ◁ G | |
756 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)(4,17,5,10)(6,11)(7,12)(8,14,25,19)(9,18,26,20)(15,29)(16,27)(21,23)(22,28)(24,30,32,31)(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)(4,14)(5,19)(6,21)(7,16)(9,23)(10,27)(11,24)(12,25)(13,17)(15,22)(18,31)(20,30)(26,29)(28,32);(1,4,7,5)(2,9,12,26)(3,14,16,19)(6,20,22,18)(8,23,25,29)(10,28,17,11)(13,24,27,32)(15,31,21,30)3026 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 5768);