Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C22 |
| direct product | C79×#12024 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C2×D8 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C4×C8 | C23×C79 |
| socle | C22×C79 | C2×#107 |
| prime radical | C23×C79 | #107 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
226 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 119 points:
(1,2,3,6,4,5,7,8)(9,10)(11,20)(12,16)(13,26)(14,19)(15,23)(17,21)(18,25)(22,33)(24,35)(27,31)(28,36)(29,40)(30,39)(32,37)(34,38)(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);(9,11,21,36,12,22,27,39,14,24,37,40,25,38,13,23)(10,15,26,34,18,29,32,35,19,30,31,33,16,28,17,20);(2,5)(6,8)(9,12,14,25)(10,16,19,18)(11,22,24,38)(13,21,27,37)(15,28,30,29)(17,31,32,26)(20,33,35,34)(23,36,39,40)6004 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5012);