Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C89 · Schur multiplier (H2) = C29 |
| direct product | C2×C89×#2951 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C89 | C22×D4 |
| commutator | C4 | C24×C4×C89 |
| Frattini, cosocle | C2×C4 | C25×C89 |
| prime radical | C22×#135×C89 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C2×#117×C89 ◁ G | |
2806 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 5 generators on 111 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(3,4,6,5)(7,8)(9,14)(10,16)(11,15)(12,17)(13,18)(19,22)(20,21);(7,9)(8,13)(10,18)(11,20)(12,19)(14,15)(16,22)(17,21);(3,5,6,4)(7,9)(8,14)(10,20)(11,18)(12,19)(13,15)(16,21)(17,22);(7,10,12,11)(8,15,17,16)(9,18,19,20)(13,14,21,22)7832 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53153);