Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C89 · Schur multiplier (H2) = C28 |
| direct product | C89×#60999 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C89 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C89 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C4×#41×C89 ◁ G | |
1682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 11392 |
A minimal faithful permutation representation — 5 generators on 137 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,31)(21,26)(22,35)(23,27)(25,32)(28,43)(29,38)(30,39)(33,41)(34,40)(36,42)(37,46)(44,47)(45,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,23,30)(18,24,27,39)(20,28,33,45)(21,34,22,29)(25,37,42,47)(26,40,35,38)(31,43,41,48)(32,46,36,44);(3,10)(6,9)(11,16)(13,15)(17,20,23,33)(18,25,27,42)(19,28,30,45)(21,32,22,36)(24,37,39,47)(26,41,35,31)(29,44,34,46)(38,43,40,48);(2,7)(6,9)(8,14)(13,15)(17,21,23,22)(18,26,27,35)(19,29,30,34)(20,32,33,36)(24,38,39,40)(25,41,42,31)(28,44,45,46)(37,43,47,48);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(19,30)(24,39)(28,45)(29,34)(37,47)(38,40)(43,48)(44,46)4361 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53987);