Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C79 · Schur multiplier (H2) = C1 |
| direct product | C79×#7544 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C79 | D4 |
| commutator | C4 | C2×C32×C79 |
| Frattini, cosocle | C2×C32 | C22×C79 |
| socle | C2×C79 | C4⋊C32 |
| prime radical | C22×C79 | M64 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C79 ◁ C2×C32×C79 ◁ G | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 143 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 2 generators on 143 points:
(1,2,5,10,6,11,22,33,3,12,16,34,17,37,42,57,4,13,19,35,20,38,45,58,15,8,25,26,40,27,48,50,7,14,23,36,24,39,47,59,18,28,43,51,44,53,61,63,21,9,30,31,46,32,55,56,41,29,49,52,60,54,62,64)(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);(1,3,4,15,7,18,21,41)(2,8,9,12,14,29,13,28)(5,16,19,25,23,43,30,49)(6,17,20,40,24,44,46,60)(10,26,31,34,36,52,35,51)(11,27,32,37,39,54,38,53)(22,42,45,48,47,61,55,62)(33,50,56,57,59,64,58,63)8848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 532);