Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C89 · Schur multiplier (H2) = C28 |
| direct product | C89×#60319 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | C25 |
| commutator | C22 | C24×C4×C89 |
| Frattini, cosocle | C2×C4 | C25×C89 |
| socle | C22×C89 | C24×C4 |
| prime radical | C89×#3460 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ G | |
1848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 5 generators on 113 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,13,15,18,19,23)(11,16,12,17,20,22,21,24)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,14)(13,17)(15,20)(18,22)(19,21)(23,24);(3,8)(5,7)(9,12)(10,17)(11,19)(13,22)(14,20)(15,21)(16,23)(18,24);(2,6)(5,7)(11,20)(12,21)(16,22)(17,24);(9,13,14,18,15,23,19,10)(11,17,12,22,20,24,21,16)6764 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53307);