Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C23 |
| direct product | C89×#27042 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C89 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| prime radical | C24×C89 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C4×C22⋊C4×C89 ◁ G | |
678 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,21,24)(19,22,27,31)(20,23,28,32)(25,29,26,30)(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);(2,6)(7,8)(10,13)(14,15)(17,19,21,27)(18,22,24,31)(20,25,28,26)(23,29,32,30);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(19,25)(20,28)(22,29)(23,32)(26,27)(30,31);(1,4,5,3)(2,7,6,8)(10,13)(14,15)4628 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 20030);