Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#12995 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #447 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | #1874 |
| prime radical | C79×#3067 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ C2×#131×C79 ◁ G | |
476 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 143 points:
(1,2,7,14,8,15,32,46,9,16,31,45,6,13,5,12)(3,17,21,39,22,43,19,36,23,50,29,10,20,37,57,38)(4,24,27,44,26,42,30,48,28,47,34,40,33,11,25,41)(18,51,54,62,53,61,52,63,55,64,59,60,58,49,56,35)(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);(1,3,8,20,9,23,6,22)(2,10,15,36,16,39,13,38)(4,18,26,58,28,55,33,53)(5,29,7,19,32,21,31,57)(11,35,24,60,42,63,47,62)(12,43,14,17,46,37,45,50)(25,59,27,52,30,54,34,56)(40,64,41,61,44,51,48,49);(1,4)(2,11)(3,18)(5,30)(6,33)(7,34)(8,26)(9,28)(10,35)(12,44)(13,47)(14,48)(15,24)(16,42)(17,49)(19,56)(20,58)(21,59)(22,53)(23,55)(25,32)(27,31)(29,54)(36,60)(37,64)(38,62)(39,63)(40,46)(41,45)(43,51)(50,61)(52,57)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5983);