Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#33540 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C79 | C2×#447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C79 ◁ C22×C4×C79 ◁ C2×#131×C79 ◁ G | |
1078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 143 points:
(1,2)(3,15)(4,20)(5,24)(6,14)(7,13)(8,11)(9,17)(10,22)(12,27)(16,43)(18,49)(19,42)(21,52)(23,39)(25,40)(26,36)(28,38)(29,34)(30,37)(31,58)(32,48)(33,56)(35,55)(41,45)(44,50)(46,63)(47,61)(51,54)(53,59)(57,62)(60,64)(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);(1,3,8,19)(2,9,11,32)(4,16,23,47)(5,25,28,56)(6,18,7,30)(10,31,34,59)(12,37,36,49)(13,33,14,40)(15,38,42,24)(17,26,48,27)(20,50,39,60)(21,53,55,58)(22,45,29,46)(35,61,52,43)(41,62,63,51)(44,57,64,54);(1,4)(2,10)(3,16)(5,21)(6,29)(7,22)(8,23)(9,31)(11,34)(12,35)(13,39)(14,20)(15,41)(17,44)(18,46)(19,47)(24,51)(25,53)(26,57)(27,54)(28,55)(30,45)(32,59)(33,60)(36,52)(37,61)(38,62)(40,50)(42,63)(43,49)(48,64)(56,58);(2,11)(9,32)(10,34)(12,36)(13,14)(15,42)(20,39)(24,38)(31,59)(33,40)(35,52)(37,49)(41,63)(43,61)(50,60)(51,62)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26528);