Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C89 · Schur multiplier (H2) = C22×C4 |
| direct product | C89×#7850 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C89 | C23 |
| commutator | C23 | C2×C42×C89 |
| Frattini, cosocle | C2×C42 | C23×C89 |
| socle | C23×C89 | C2×C42 |
| prime radical | C24×C89 | C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C89 ◁ G | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 3 generators on 117 points:
(1,2)(3,4)(5,6,8,12,10,11,9,7)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28)(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);(2,4)(5,7,9,11,10,12,8,6)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(6,11)(7,12)(15,21)(18,24)(22,27)(25,28)7832 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 838);