Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#12980 |
| 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×#135×C79 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C79 ◁ C22×C16×C79 ◁ G | |
470 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)(5,6,10,15,11,16,30,34,12,17,31,35,9,14,19,20)(7,18,22,36,21,26,33,29,23,28,13,32,27,8,24,25)(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);(2,4)(5,7,19,33,9,27,31,22,12,23,30,24,11,21,10,13)(6,8,20,36,14,28,35,25,17,26,34,32,16,18,15,29);(5,8)(6,13)(7,20)(9,28)(10,29)(11,18)(12,26)(14,33)(15,21)(16,24)(17,22)(19,36)(23,34)(25,31)(27,35)(30,32)6004 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5968);