Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7540 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C79 · Schur multiplier (H2) = C2 |
| direct product | C79×#7540 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | #1415 |
| commutator | C32 | C2×C4×C79 |
| Frattini, cosocle | C2×C32 | C22×C79 |
| prime radical | C79×#2262 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C2×C4×C79 ◁ C2×C8×C79 ◁ C2×C16×C79 ◁ C2×C32×C79 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 2 generators on 143 points:
(2,8,9,30)(3,13)(4,19)(5,21)(6,25)(10,36,34,44)(11,38,35,43)(12,39,27,33)(14,20,31,29)(15,23)(16,48)(17,52)(18,28)(22,47)(24,32,60,62)(26,58)(37,49,56,64)(40,41)(42,50)(45,59,53,57)(46,61)(51,63)(54,55)(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);(1,2)(3,14)(4,20)(5,24)(6,27)(7,9)(8,28)(10,19)(11,21)(12,25)(13,30)(15,34)(16,49)(17,53)(18,31)(22,59)(23,29)(26,60)(32,63)(33,42)(35,58)(36,61)(37,47)(38,46)(39,52)(40,44)(41,43)(45,50)(48,62)(51,64)(54,57)(55,56)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 528);