Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#21478 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | C2×#120 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C2×C79 | C4×#120 |
| prime radical | C79×#3018 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C8×C79 ◁ G | |
1282 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,7,11,8,12,6,20)(3,13,16,9,17,27,23,26)(4,10,18,22,19,21,24,5)(14,29,31,32,30,15,28,25)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(1,3)(2,9)(4,14)(5,15)(6,23)(7,16)(8,17)(10,25)(11,27)(12,26)(13,20)(18,31)(19,30)(21,32)(22,29)(24,28);(1,4)(2,10)(3,14)(5,20)(6,24)(7,18)(8,19)(9,25)(11,22)(12,21)(13,15)(16,31)(17,30)(23,28)(26,32)(27,29);(1,5,7,10,8,22,6,21)(2,4,11,18,12,19,20,24)(3,15,16,25,17,29,23,32)(9,14,27,31,26,30,13,28)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 14466);