Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C23×C4 |
| direct product | C89×#19742 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C89 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G | |
738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 4 generators on 121 points:
(1,2,7,4)(3,10,13,16)(5,9,6,14)(8,15,11,12)(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);(1,3,6,12,7,13,5,15)(2,8,9,10,4,11,14,16)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(1,4,7,2)(3,11,13,8)(5,9,6,14)(10,12,16,15)(17,19,20,25,21,26,27,31)(18,22,24,28,23,29,30,32);(1,5,7,6)(2,9,4,14)(3,12,13,15)(8,16,11,10)(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, 12730);