Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#13125 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C79 | C2×D4 |
| commutator | C2×C4 | C22×C8×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C22×C79 | C8×D4 |
| prime radical | C2×#41×C79 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C79 ◁ C22×C16×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)(5,6,8,14,9,15,18,12,10,16,23,32,24,13,33,26)(7,17,21,27,20,11,25,28,22,29,19,31,36,30,35,34)(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);(5,7,18,25,24,36,8,21,10,22,33,35,9,20,23,19)(6,11,26,27,13,17,32,34,16,30,12,31,15,29,14,28);(2,4)(5,8,9,18,10,23,24,33)(6,12,13,14,16,26,15,32)(7,19,20,35,22,21,36,25)(11,27,17,34,30,31,29,28)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6113);