Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C89 · Schur multiplier (H2) = C2×C4 |
| direct product | C89×#11634 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C89 |
| Frattini, cosocle | C4×C8 | C23×C89 |
| socle | C22×C89 | C2×C22⋊C8 |
| prime radical | C22×D4×C89 | C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ C2×C4×C8×C89 ◁ G | |
386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 3 generators on 113 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,17,14,18,21,24,13,15,20,22,11,16,19,23)(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);(2,6)(3,4)(7,8)(10,15)(12,20)(16,18)(19,21);(1,3,5,4)(2,7,8,6)(9,11,13,14)(10,16,15,18)(12,19,20,21)(17,23,22,24)5696 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 4622);