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 | SD128×C157 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C157 | D32 |
| commutator, Frattini, cosocle | C32 | C22×C157 |
| prime radical | D32×C157 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C157 ◁ C4×C157 ◁ C8×C157 ◁ C16×C157 ◁ C32×C157 ◁ G | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 221 |
| transitive degree | μt(G) = 10048 |
A minimal faithful permutation representation — 2 generators on 221 points:
(1,2,7,13)(3,8,18,30)(4,19,22,53)(5,23,25,56)(6,26,24,12)(9,31,34,14)(10,35,37,15)(11,38,40,20)(16,39,49,64)(17,41,48,29)(21,54,45,52)(27,58,60,43)(28,61,62,42)(32,63,59,47)(33,44,57,50)(36,46,51,55)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221);(2,8)(3,14)(4,15)(5,20)(6,24)(9,10)(11,39)(12,41)(13,30)(16,47)(17,50)(18,31)(19,27)(21,55)(22,35)(23,28)(25,38)(26,29)(32,52)(33,51)(34,37)(36,57)(40,64)(42,43)(44,48)(45,46)(49,63)(53,60)(54,59)(56,62)(58,61)5495 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20096, 162);