Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C25 |
| direct product | C89×#32961 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C89 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C89 |
| prime radical | C22×#41×C89 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C22×C42×C89 ◁ G | |
1320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 4 generators on 117 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14,17,19)(15,22,24,28)(16,21,18,25)(20,26,27,23)(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);(5,7,8,12)(6,9,10,11)(14,19)(20,27)(21,25)(22,28);(13,15)(14,20)(16,23)(17,24)(18,26)(19,27)(21,28)(22,25);(2,4)(13,16,17,18)(14,21,19,25)(15,23,24,26)(20,28,27,22)5162 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 25949);