Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#29387 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C89 | #535 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| socle | C2×C89 | #2598 |
| prime radical | C89×#2017 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C89 ◁ C22×C4×C89 ◁ G | |
646 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)(3,13)(4,17)(5,10)(6,11)(7,12)(8,15)(9,19)(14,22)(16,23)(18,27)(20,24)(21,26)(25,29)(28,31)(30,32)(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);(1,3,5,15,7,16,6,20)(2,8,11,13,12,24,10,23)(4,14,19,30,18,29,21,31)(9,25,17,32,26,22,27,28);(1,4)(2,9)(3,14)(5,19)(6,21)(7,18)(8,22)(10,27)(11,17)(12,26)(13,28)(15,30)(16,29)(20,31)(23,32)(24,25);(2,10,12,11)(4,18)(8,23,24,13)(9,17,26,27)(14,29)(19,21)(22,28,25,32)(30,31)3560 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 22375);