Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#11597 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #102 |
| commutator | C8 | C22×C8×C79 |
| Frattini, cosocle | C4×C8 | C23×C79 |
| socle | C22×C79 | C2×#102 |
| prime radical | C2×#135×C79 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 119 points:
(1,2,3,5,4,6,7,8)(9,10)(11,20,23,37)(12,17)(13,25)(14,29)(15,19)(16,22,32,40)(18,27)(21,34,30,38)(24,35)(26,31,39,33)(28,36)(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)(10,16)(12,21)(13,26)(14,30)(15,23)(17,31)(18,34)(19,32)(20,36)(22,24)(25,37)(27,40)(28,39)(29,33)(35,38);(9,12,15,14)(10,17,19,29)(11,21,23,30)(13,24,28,27)(16,31,32,33)(18,25,35,36)(20,34,37,38)(22,39,40,26)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 4585);