Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33752 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C23×C4 |
| direct product | C79×#33776 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| prime radical | C79×#3147 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
592 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 119 points:
(1,2)(3,7)(4,6)(5,8)(9,10,14,19)(11,20,24,38)(12,16,26,18)(13,27,29,33)(15,30,32,22)(17,34,36,25)(21,35,23,40)(28,37,39,31)(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);(1,3,4,8)(2,5,6,7);(3,8)(5,7)(9,11)(10,15)(12,21)(13,28)(14,24)(16,31)(17,35)(18,37)(19,32)(20,27)(22,25)(23,26)(29,39)(30,34)(33,38)(36,40);(2,6)(5,7)(9,12,14,26)(10,16,19,18)(11,21,24,23)(13,25,29,34)(15,31,32,37)(17,33,36,27)(20,35,38,40)(22,39,30,28)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26764);