Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C89 · Schur multiplier (H2) = C23 |
| direct product | C2×C89×#1353 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C89 | #99 |
| commutator | C22×C4 | C22×C4×C89 |
| Frattini, cosocle | C2×C4⋊C4 | C23×C89 |
| prime radical | C24×C89 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C89 ◁ C24×C89 ◁ C22×C4⋊C4×C89 ◁ G | |
500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5696 |
A minimal faithful permutation representation — 3 generators on 115 points:
(1,2)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(4,7,8,9)(11,12,14,18,15,17,21,24)(13,16,20,22,19,23,26,25);(3,4)(5,9)(6,8)(7,10)(11,13)(12,16,17,23)(14,20)(15,19)(18,25,24,22)(21,26)4094 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 4707);