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 |
| direct product | C79×#13164 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C79 | #121 |
| commutator | C2×C4 | C22×C8×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | #1939 |
| prime radical | C23×C79 | #131 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C79 ◁ C22×C8×C79 ◁ G | |
284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,4,12,6,10,19,9,7,13,20,27,5,11,18,28)(3,14,15,29,16,8,22,24,17,25,30,23,21,26,31,32)(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);(1,3,4,15,6,16,19,22,7,17,20,30,5,21,18,31)(2,8,12,24,10,25,9,23,13,26,27,32,11,14,28,29);(1,4,6,19,7,20,5,18)(2,9,11,12,13,28,10,27)(3,15,16,22,17,30,21,31)(8,23,14,24,26,29,25,32)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6152);