Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#13289 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#102 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C23×C79 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 119 points:
(1,2,4,6)(3,7,8,5)(9,10,14,18,15,19,13,29)(11,20,23,38,24,33,30,37)(12,25,27,32,28,40,26,17)(16,31,35,22,36,39,34,21)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(9,11,12,21,13,23,26,31,15,24,28,22,14,30,27,39)(10,16,32,20,18,34,40,37,19,36,17,33,29,35,25,38);(1,3,4,8)(2,5,6,7)(9,12,13,26,15,28,14,27)(10,17,18,25,19,32,29,40)(11,21,23,31,24,22,30,39)(16,33,34,38,36,20,35,37)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6277);