Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C89 · Schur multiplier (H2) = C23×C8 |
| direct product | C89×#19721 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24×C89 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C2×C42×C89 ◁ G | |
798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 113 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 4 generators on 113 points:
(1,2,7,3)(4,5,8,6)(9,10,15,19)(11,20,23,24)(12,17,16,22)(13,18,14,21)(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);(1,3,5,8,7,2,6,4)(9,11,14,17,15,23,13,22)(10,16,18,20,19,12,21,24);(1,4,6,2,7,8,5,3)(9,12)(10,17)(11,21)(13,20)(14,24)(15,16)(18,23)(19,22);(1,5,7,6)(2,8,3,4)(9,13,15,14)(10,18,19,21)(11,22,23,17)(12,24,16,20)3560 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 12709);