Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11704 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#11926 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#102 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C4×C8 | C23×C79 |
| prime radical | C22×D8×C79 | C4 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
418 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 119 points:
(1,2,3,6,4,5,7,8)(9,10)(11,21,24,37)(12,17)(13,27)(14,19)(15,20)(16,23,32,40)(18,26)(22,35,25,38)(28,31,39,33)(29,36)(30,34)(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);(2,5)(3,7)(9,11)(10,16)(12,25)(13,28)(14,22)(15,24)(17,33)(18,35)(19,31)(20,32)(21,34)(23,29)(26,40)(27,37)(30,39)(36,38);(9,12,15,14)(10,17,20,19)(11,22,24,25)(13,26,30,29)(16,31,32,33)(18,34,36,27)(21,35,37,38)(23,39,40,28)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 4914);