Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C22×C4 |
| direct product | C79×#13245 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #466 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C22×D16×C79 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ #405×C79 ◁ G | |
442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 115 points:
(1,2,4,3)(5,6,11,17,13,18,12,7)(8,22,25,35,27,34,26,31)(9,16,10,29,19,21,28,15)(14,30,32,36,33,23,20,24)(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);(1,3,4,2)(5,7,19,21,12,6,10,15,13,17,9,16,11,18,28,29)(8,20,30,34,26,14,24,31,27,32,23,22,25,33,36,35);(5,8,9,23,12,26,28,36,13,27,19,30,11,25,10,24)(6,14,29,35,17,32,21,34,18,33,15,31,7,20,16,22)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6233);