Finite Group DatabasebetaFinite Group Databasebeta| order | 20096 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C157 · Schur multiplier (H2) = C1 |
| direct product | Q128×C157 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C157 | D32 |
| commutator, Frattini, cosocle | C32 | C22×C157 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C157 ◁ C4×C157 ◁ C8×C157 ◁ C16×C157 ◁ C32×C157 ◁ G | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 285 |
| transitive degree | μt(G) = 20096 |
A minimal faithful permutation representation — 2 generators on 285 points:
(1,2,8,14)(3,15,20,60)(4,21,27,57)(5,28,32,89)(6,33,36,95)(7,37,35,13)(9,39,43,16)(10,44,48,24)(11,49,52,23)(12,53,55,30)(17,63,65,124)(18,66,67,126)(19,68,38,59)(22,50,75,117)(25,81,84,122)(26,85,83,71)(29,45,92,109)(31,93,77,88)(34,58,96,123)(40,100,102,73)(41,103,104,72)(42,62,56,98)(46,110,112,127)(47,80,97,107)(51,78,87,114)(54,99,119,61)(64,125,90,111)(69,106,120,76)(70,105,121,79)(74,118,86,116)(82,115,128,101)(91,113,94,108)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285);(1,3,8,20)(2,9,14,43)(4,22,27,75)(5,29,32,92)(6,34,36,96)(7,38,35,19)(10,45,48,109)(11,50,52,117)(12,54,55,119)(13,56,37,42)(15,57,60,21)(16,44,39,24)(17,49,65,23)(18,53,67,30)(25,82,84,128)(26,86,83,74)(28,40,89,102)(31,94,77,91)(33,41,95,104)(46,111,112,125)(47,113,97,108)(51,118,87,116)(58,122,123,81)(59,71,68,85)(61,110,99,127)(62,80,98,107)(63,120,124,69)(64,78,90,114)(66,121,126,70)(72,106,103,76)(73,105,100,79)(88,115,93,101)5495 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20096, 163);