Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C22 |
| direct product | C79×#13638 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #367 |
| commutator | C16 | C22×C4×C79 |
| Frattini, cosocle | C2×C16 | C23×C79 |
| socle | C2×C79 | C2×#367 |
| prime radical | C23×C79 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ C22×C8×C79 ◁ C22×C16×C79 ◁ G | |
182 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)(3,14,18,41)(4,10)(5,23)(6,27)(7,30)(8,13)(9,17,33,49)(11,24)(12,28)(15,43)(16,39,20,55)(19,53,52,62)(21,31)(22,38)(25,57)(26,56)(29,36)(32,47,59,51)(34,44,60,64)(35,46)(37,45)(40,61,58,50)(42,48,63,54)(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);(1,3,15,44,6,19,5,17,4,16,24,50,22,48,21,47,8,18,46,64,29,52,26,49,7,20,45,61,28,54,25,51)(2,9,31,53,12,34,11,14,30,59,23,42,36,40,43,55,13,33,57,62,38,60,37,41,10,32,56,63,27,58,35,39);(1,4,8,7)(2,10,13,30)(3,16,18,20)(5,21,26,25)(6,22,29,28)(9,32,33,59)(11,35,37,43)(12,27,38,36)(14,39,41,55)(15,24,46,45)(17,47,49,51)(19,48,52,54)(23,31,56,57)(34,58,60,40)(42,53,63,62)(44,50,64,61)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6626);