Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C89 · Schur multiplier (H2) = C24 |
| direct product | C89×#33426 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C89 | C2×#117 |
| commutator | C2×C4 | C23×C4×C89 |
| Frattini, cosocle | C2×C8 | C24×C89 |
| socle | C2×C89 | C2×#432 |
| prime radical | C2×#41×C89 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C89 ◁ C2×#97×C89 ◁ G | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 2848 |
A minimal faithful permutation representation — 4 generators on 121 points:
(1,2,4,10,5,11,18,27,6,9,17,25,19,8,23,24)(3,12,14,26,15,20,29,31,13,7,16,21,28,22,30,32)(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)(2,7)(4,16)(5,15)(6,13)(8,20)(9,12)(10,26)(11,22)(14,17)(18,30)(19,28)(21,25)(23,29)(24,32)(27,31);(2,8,9,11)(4,17)(7,20,12,22)(10,27,25,24)(14,16)(18,23)(21,32,26,31)(29,30);(3,13)(7,12)(14,16)(15,28)(20,22)(21,26)(29,30)(31,32)5162 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 26414);