Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C89 · Schur multiplier (H2) = C23×C4 |
| direct product | C89×#22815 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C89 | C22×D4 |
| commutator | C2×C4 | C23×C4×C89 |
| Frattini, cosocle | C22×C4 | C24×C89 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C4×C4⋊C4×C89 ◁ G | |
754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 22784 |
A minimal faithful permutation representation — 4 generators on 117 points:
(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,14,18,17,19,22,16,21)(15,23,24,20,26,27,25,28)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117);(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,24)(17,23)(18,25)(19,26)(21,27)(22,28);(5,8,9,7)(6,10,11,12)(13,16,19,18)(14,21,22,17)(15,24,26,25)(20,27,28,23);(13,17,18,22,19,21,16,14)(15,20,24,27,26,28,25,23)5162 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 15803);