Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13005 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#13006 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | #1874 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C2×#41×C79 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C22×C4×C79 ◁ C2×#131×C79 ◁ G | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 143 points:
(1,2,7,13,32,35,30,11,8,14,29,48,6,12,5,27)(3,15,19,9,31,38,17,46,20,50,59,39,18,44,33,37)(4,21,24,40,23,43,28,47,26,42,34,45,25,10,22,41)(16,51,54,60,53,61,52,36,56,63,58,62,55,49,57,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143);(1,3,6,31,8,20,32,18)(2,9,35,37,14,39,12,46)(4,16,25,53,26,56,23,55)(5,19,29,17,30,59,7,33)(10,36,21,60,43,64,42,62)(11,44,48,50,27,38,13,15)(22,54,34,52,28,58,24,57)(40,51,47,49,45,63,41,61);(1,4)(2,10)(3,16)(5,28)(6,25)(7,34)(8,26)(9,36)(11,45)(12,42)(13,47)(14,43)(15,49)(17,57)(18,55)(19,58)(20,56)(21,35)(22,30)(23,32)(24,29)(27,40)(31,53)(33,52)(37,60)(38,51)(39,64)(41,48)(44,63)(46,62)(50,61)(54,59)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5994);