Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#19161 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | #121 |
| commutator, Frattini, cosocle | C42 | C24×C79 |
| socle | C22×C79 | C2×#121 |
| prime radical | C2×#41×C79 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C2×C42×C79 ◁ G | |
598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,5,9)(3,10,12,16)(4,7,11,8)(6,14,15,13)(17,18)(19,26)(20,23)(21,25)(22,28)(24,30)(27,31)(29,32)(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);(1,3,11,14,5,12,4,13)(2,6,7,16,9,15,8,10);(3,12)(6,15)(10,16)(13,14)(17,19)(18,22)(20,27)(21,29)(23,26)(24,32)(25,31)(28,30);(1,4,5,11)(2,7,9,8)(3,13,12,14)(6,16,15,10)(17,20,21,30)(18,23,25,24)(19,27,29,28)(22,26,31,32)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 12149);