Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#13625 |
| 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 | |
290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 143 points:
(1,2,6,12,8,14,4,10)(3,15,16,46,20,49,19,57)(5,23,26,61,28,45,21,58)(7,29,25,44,31,37,22,13)(9,32,33,17,35,50,42,54)(11,38,40,52,41,60,36,27)(18,55,51,39,56,64,30,48)(24,43,59,63,53,47,62,34)(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)(2,9)(4,16)(5,24)(6,19)(7,30)(8,20)(10,33)(11,39)(12,42)(13,43)(14,35)(15,45)(17,52)(18,22)(21,59)(23,49)(25,56)(26,62)(27,54)(28,53)(29,34)(31,51)(32,60)(36,64)(37,63)(38,50)(40,55)(41,48)(44,47)(46,61)(57,58);(1,4,8,6)(2,10,14,12)(3,16,20,19)(5,21,28,26)(7,22,31,25)(9,33,35,42)(11,36,41,40)(13,37,44,29)(15,46,49,57)(17,50,54,32)(18,51,56,30)(23,58,45,61)(24,59,53,62)(27,60,52,38)(34,43,63,47)(39,64,48,55)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6613);