Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#18515 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #120 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| socle | C22×C79 | C2×#120 |
| prime radical | C79×#3483 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C42×C79 ◁ G | |
976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2)(3,10)(4,7)(5,9)(6,12)(8,14)(11,15)(13,16)(17,18,21,24)(19,25,26,32)(20,28,29,23)(22,30,31,27)(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)(2,6)(4,11)(5,13)(7,10)(8,16)(9,15)(12,14)(20,29)(23,28)(25,32)(27,30);(17,19,21,26)(18,22,24,31)(20,30,29,27)(23,32,28,25);(1,4,5,14)(2,7,9,8)(3,11,13,12)(6,10,15,16)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 11503);