Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C89 · Schur multiplier (H2) = C28 |
| direct product | C4×#563×C89 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C89 | C24 |
| commutator | C22 | C24×C4×C89 |
| Frattini, cosocle | C23 | C25×C89 |
| socle | C23×C89 | C25 |
| prime radical | C22×D4×C89 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C89 ◁ G | |
2024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 5 generators on 117 points:
(1,2,3,4)(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);(5,6)(7,9)(8,10)(11,12)(13,14,17,20)(15,21,23,18)(16,19,25,28)(22,27,24,26);(5,7)(6,9)(8,11)(10,12)(13,15,17,23)(14,18,20,21)(16,24,25,22)(19,27,28,26);(6,10)(7,11)(13,16,17,25)(14,19,20,28)(15,22,23,24)(18,26,21,27);(6,10)(9,12)(14,20)(15,23)(19,28)(22,24)6764 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 27342);