Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C29 |
| direct product | C2×C89×#3273 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C89 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C89 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C2×C4×D4×C89 ◁ G | |
3042 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 5696 |
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)(5,6)(7,8,11,14)(9,15,16,22)(10,19,20,13)(12,17,21,18);(7,9)(8,12)(10,17)(11,16)(13,22)(14,21)(15,19)(18,20);(4,6)(9,16)(12,21)(15,22)(17,18);(9,17)(10,20)(12,22)(13,19)(15,21)(16,18)5162 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 54020);