Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#22690 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | D42 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C2×#135×C79 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ G | |
846 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)(7,17)(8,23)(9,14)(10,28)(11,16)(12,20)(13,24)(15,29)(18,30)(19,36)(21,32)(22,31)(25,33)(26,34)(27,35)(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);(5,7,11,22)(6,12,16,32)(8,18,19,26)(9,25,27,24)(10,21,29,20)(13,30,33,34)(14,23,35,36)(15,31,28,17);(5,8,10,24,11,19,29,25)(6,13,15,23,16,33,28,36)(7,18,21,9,22,26,20,27)(12,30,31,35,32,34,17,14);(5,9)(6,14)(7,19)(8,22)(10,26)(11,27)(12,13)(15,30)(16,35)(17,36)(18,29)(20,24)(21,25)(23,31)(28,34)(32,33)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 15678);