Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C25 |
| direct product | C89×#12718 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C89 | #1781 |
| commutator | C22×C4 | C22×C4×C89 |
| Frattini, cosocle | C2×#41 | C23×C89 |
| prime radical | C89×#3498 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C89 ◁ C2×C4×C89 ◁ C2×H(5,𝔽2)×C89 ◁ G | |
1080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 105 |
| transitive degree | μt(G) = 1424 |
A minimal faithful permutation representation — 3 generators on 105 points:
(1,2)(3,12)(4,14,5,9)(6,10)(7,11)(8,13,16,15)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105);(1,3,7,6)(2,8,11,16)(4,13,5,15)(9,10,14,12);(1,4,7,5)(2,8,11,16)(3,13,6,15)(9,12,14,10)3026 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 5706);