Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#33380 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C2×D8 |
| commutator | C8 | C23×C4×C79 |
| Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C22×C79 | C22×D8 |
| prime radical | C22×#135×C79 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ C23×C4×C79 ◁ C23×C8×C79 ◁ G | |
1170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 115 points:
(1,2,3,4)(5,6)(7,16)(8,13)(9,23)(10,12)(11,18)(14,21)(15,22)(17,30)(19,35)(20,32)(24,31)(25,29)(26,34)(27,33)(28,36)(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)(6,11)(8,17)(9,24)(10,20)(12,26)(13,27)(14,30)(15,31)(16,23)(18,21)(19,22)(25,33)(28,35)(29,32)(34,36);(6,12)(11,26)(13,15)(14,28)(16,32)(23,29)(27,31)(30,35);(5,8,10,22)(6,13,12,15)(7,17,20,19)(9,21,25,36)(11,27,26,31)(14,29,28,23)(16,30,32,35)(18,33,34,24)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26368);