Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#12617 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C79 | #117 |
| commutator | C2×C4 | C2×C42×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | #1752 |
| prime radical | C2×#41×C79 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C79 ◁ C2×M32×C79 ◁ G | |
382 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 143 points:
(1,2,6,12,7,13,28,42,8,14,29,43,5,11,26,40)(3,15,18,44,17,33,47,41,19,39,27,55,25,9,30,32)(4,10,22,36,24,38,54,63,23,37,53,62,21,35,51,60)(16,34,45,58,20,48,49,64,46,59,56,61,50,31,52,57)(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)(4,20,23,50)(5,25)(6,27)(7,17)(8,19)(10,34,37,59)(11,39)(12,41)(13,15)(14,33)(16,24,46,21)(18,29)(22,52,53,49)(26,47)(28,30)(31,38,48,35)(32,43)(36,61,62,58)(40,44)(42,55)(45,51,56,54)(57,60,64,63);(1,4,6,22,7,24,28,54,8,23,29,53,5,21,26,51)(2,10,12,36,13,38,42,63,14,37,43,62,11,35,40,60)(3,16,18,45,17,20,47,49,19,46,27,56,25,50,30,52)(9,31,32,57,15,34,44,58,33,48,41,64,39,59,55,61)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5605);