Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C23×C4 |
| direct product | C89×#19737 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C89 |
| prime radical | C89×#3205 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G | |
746 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 4 generators on 121 points:
(2,6)(4,11)(7,12)(8,13)(10,15)(14,16)(19,25)(20,27)(22,28)(24,30)(26,31)(29,32)(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);(1,2,4,6,5,8,11,13)(3,7,10,12,9,14,15,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(1,3,5,9)(2,7,8,14)(4,10,11,15)(6,12,13,16)(17,19,20,25,21,26,27,31)(18,22,24,28,23,29,30,32);(2,8)(3,9)(6,13)(10,15)(18,23)(22,29)(24,30)(28,32)3560 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 12725);