Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#13152 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C2×D8 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C22×C79 | C4×D8 |
| prime radical | C2×#135×C79 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C22×C8×C79 ◁ C2×C4×C8×C79 ◁ G | |
324 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 115 points:
(1,2)(3,4)(5,6,10,16,11,18,26,17)(7,19,23,30,24,34,36,33)(8,25,27,14,20,15,9,28)(12,29,31,35,32,22,13,21)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(5,7,20,35,26,36,27,29,11,24,8,21,10,23,9,22)(6,12,14,19,16,31,15,30,18,32,28,34,17,13,25,33);(2,4)(5,7,8,21,26,36,9,22,11,24,20,35,10,23,27,29)(6,13,14,33,16,12,15,19,18,31,28,30,17,32,25,34)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6140);