Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C4 |
| direct product | C79×#13113 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C2×D8 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C22×C79 | C4×D8 |
| prime radical | C23×C79 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C22×C8×C79 ◁ C2×C4×C8×C79 ◁ G | |
262 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)(9,10,15,21,13,19,16,31)(11,22,24,20,14,17,25,33)(12,26,29,40,27,38,32,37)(18,30,35,39,34,23,36,28)(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)(6,8)(9,11,15,24,13,14,16,25)(10,17,21,33,19,22,31,20)(12,23,29,28,27,30,32,39)(18,26,35,40,34,38,36,37);(1,3,4,7)(2,6,5,8)(9,12,13,27)(10,18,19,34)(11,23,14,30)(15,29,16,32)(17,26,22,38)(20,37,33,40)(21,35,31,36)(24,28,25,39)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6101);