Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C22 |
| direct product | C79×#12667 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C79 | #117 |
| commutator | C2×C4 | C2×C42×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | #1770 |
| prime radical | C23×C79 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C79 ◁ C2×M32×C79 ◁ G | |
296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,6,11,7,12,21,27,5,10,18,26,20,4,9,19)(3,8,14,23,15,24,29,32,16,13,22,28,30,17,25,31)(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)(2,8,10,13)(4,17,12,24)(5,16)(6,14)(7,15)(9,25)(11,23,26,28)(18,22)(19,31,27,32)(20,30)(21,29);(1,4,18,27,7,2,9,26,5,12,6,19,20,10,21,11)(3,13,14,28,15,17,29,31,16,8,22,23,30,24,25,32)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5655);