Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#13622 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #357 |
| commutator | C16 | C22×C4×C79 |
| Frattini, cosocle | C2×C16 | C23×C79 |
| socle | C2×C79 | C2×#357 |
| prime radical | C2×#508×C79 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ C22×C8×C79 ◁ C22×C16×C79 ◁ G | |
294 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)(3,14,17,45)(4,9)(5,23)(6,28)(7,13)(8,16,32,50)(10,26)(11,21)(12,22)(15,43,19,55)(18,54,53,64)(20,56)(24,39,51,44)(25,40)(27,42)(29,46,48,36)(30,35)(31,47,41,52)(33,57,63,61)(34,58)(37,62)(38,49)(59,60)(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)(2,8)(4,15)(5,24)(6,29)(7,17)(9,31)(10,36)(11,39)(12,41)(13,32)(14,42)(16,49)(18,21)(19,22)(20,57)(23,45)(25,53)(26,50)(27,51)(28,33)(30,48)(34,64)(35,63)(37,54)(38,46)(40,44)(43,60)(47,62)(52,58)(55,56)(59,61);(1,4,7,22)(2,9,13,12)(3,15,17,19)(5,20,27,59)(6,21,30,25)(8,31,32,41)(10,34,38,37)(11,35,40,28)(14,43,45,55)(16,47,50,52)(18,48,53,29)(23,56,42,60)(24,57,51,61)(26,58,49,62)(33,39,63,44)(36,64,46,54)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6610);