Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C22 |
| direct product | C79×#13187 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #476 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C23×C79 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ C2×C4×C8×C79 ◁ G | |
294 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)(7,18,22,34)(8,23)(9,16)(10,15)(11,17)(12,20,30,35)(13,24)(14,25)(19,29,36,33)(21,28,27,32)(26,31)(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)(5,7,10,27,11,22,9,21)(6,12,16,33,17,30,15,29)(8,19,25,20,26,36,24,35)(13,28,31,34,14,32,23,18);(5,8,10,25,11,26,9,24)(6,13,16,31,17,14,15,23)(7,19,27,20,22,36,21,35)(12,28,33,34,30,32,29,18)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6175);