Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#12911 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #427 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | #1846 |
| prime radical | C23×C79 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ #117×C79 ◁ #405×C79 ◁ G | |
266 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)(3,14,18,38)(4,20,23,50)(5,25)(6,12)(7,29)(8,13)(9,16,32,45)(10,22,34,24)(11,27)(15,39)(17,37,19,47)(21,51,53,33)(26,35)(28,55)(30,57)(31,44,56,46)(36,42)(40,61)(41,59)(43,60)(48,63)(49,52,64,54)(58,62)(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);(1,3,5,16,7,19,27,46,8,18,28,45,6,17,26,44)(2,9,11,14,12,31,25,37,13,32,35,38,29,56,55,47)(4,15,21,40,24,42,54,62,23,43,53,63,22,41,52,57)(10,30,33,36,20,48,49,39,34,58,51,59,50,61,64,60);(1,4)(2,10,13,34)(3,15)(5,21)(6,22)(7,24)(8,23)(9,30,32,58)(11,33,35,51)(12,20,29,50)(14,36,38,59)(16,40)(17,41)(18,43)(19,42)(25,49,55,64)(26,52)(27,54)(28,53)(31,48,56,61)(37,39,47,60)(44,57)(45,63)(46,62)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5899);