Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#12987 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C79 | #121 |
| commutator | C2×C4 | C22×C8×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | #1869 |
| prime radical | #575×C79 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C79 ◁ C22×C8×C79 ◁ G | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 143 points:
(1,2,6,13,7,14,28,43,8,11,29,30,5,12,25,40)(3,9,18,33,17,32,49,59,16,34,26,41,23,38,54,61)(4,19,22,52,21,37,53,42,20,39,27,55,24,10,35,36)(15,44,46,62,47,31,56,58,48,51,50,64,45,57,63,60)(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);(1,3,6,26,7,17,28,54,8,16,29,18,5,23,25,49)(2,9,30,33,14,32,40,59,11,34,13,41,12,38,43,61)(4,15,27,46,21,47,35,56,20,48,22,50,24,45,53,63)(10,31,36,60,19,51,52,62,37,57,42,58,39,44,55,64);(1,4)(2,10)(3,15)(5,24)(6,27)(7,21)(8,20)(9,31)(11,37)(12,39)(13,42)(14,19)(16,48)(17,47)(18,50)(22,29)(23,45)(25,53)(26,46)(28,35)(30,36)(32,51)(33,60)(34,57)(38,44)(40,52)(41,58)(43,55)(49,63)(54,56)(59,62)(61,64)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5975);