Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16×C79 · Schur multiplier (H2) = C1 |
| direct product | C79×#7511 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C79 | C42 |
| commutator | C4 | C4×C16×C79 |
| Frattini, cosocle | C2×C32 | C22×C79 |
| socle | C2×C79 | #1397 |
| prime radical | C22×C79 | M64 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C16×C79 ◁ G | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 2 generators on 143 points:
(1,2,3,9,4,11,15,30,5,12,16,31,18,34,39,55,6,13,17,28,19,35,40,51,21,36,41,52,42,57,59,63,7,10,14,29,20,32,37,53,22,33,38,54,43,56,58,64,23,8,24,25,44,26,46,48,45,27,47,49,60,50,61,62)(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);(2,8,10,13)(3,14)(9,28,29,25)(11,26,32,35)(12,27,33,36)(15,37)(16,38)(17,24)(30,51,53,48)(31,52,54,49)(34,50,56,57)(39,58)(40,46)(41,47)(55,63,64,62)(59,61)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 499);