Finite Group DatabasebetaFinite Group Databasebeta| order | 22784 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C89 · Schur multiplier (H2) = C28 |
| direct product | C89×#60296 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C89 | C25 |
| commutator | C22 | C24×C4×C89 |
| Frattini, cosocle | C2×C4 | C25×C89 |
| socle | C22×C89 | C24×C4 |
| prime radical | C89×#3455 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C89 ◁ G | |
1684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 5696 |
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)(17,18,21,25,22,26,32,38)(19,23,29,35,28,34,41,45)(20,24,31,37,30,36,42,46)(27,33,40,44,39,43,47,48)(49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,33)(25,35)(26,34)(30,39)(31,40)(32,41)(36,43)(37,44)(38,45)(42,47)(46,48);(3,10)(6,9)(11,16)(13,15)(17,20)(18,24)(19,27)(21,31)(22,30)(23,33)(25,37)(26,36)(28,39)(29,40)(32,42)(34,43)(35,44)(38,46)(41,47)(45,48);(2,7)(6,9)(8,14)(13,15)(20,30)(24,36)(27,39)(31,42)(33,43)(37,46)(40,47)(44,48);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(19,28)(23,34)(27,39)(29,41)(33,43)(35,45)(40,47)(44,48)6764 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(22784, 53284);