Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C22×C4 |
| direct product | C79×#13197 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C79 | #121 |
| commutator | C2×C4 | C22×C8×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | #1954 |
| prime radical | C2×#135×C79 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C79 ◁ C22×C8×C79 ◁ G | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,6,12,5,10,15,25,7,13,4,20,22,11,21,9)(3,14,16,31,18,8,24,27,19,28,23,26,17,29,32,30)(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);(1,3,15,32,22,17,6,23,7,19,21,24,5,18,4,16)(2,8,25,31,11,14,12,30,13,29,9,26,10,28,20,27);(1,4,5,21,7,6,22,15)(2,9,11,20,13,25,10,12)(3,16,18,24,19,23,17,32)(8,26,14,27,29,31,28,30)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6185);