Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#22896 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | D42 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| prime radical | C2×H(5,𝔽2)×C79 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ G | |
1016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 4 generators on 115 points:
(1,2,4,3)(5,6)(7,19)(8,24)(9,15)(10,17)(11,16)(12,18)(13,22)(14,25)(20,29)(21,34)(23,33)(26,36)(27,31)(28,30)(32,35)(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);(1,3,4,2)(5,7)(6,13)(8,20)(9,26)(10,28)(11,22)(12,23)(14,29)(15,32)(16,33)(17,19)(18,30)(21,27)(24,34)(25,35)(31,36);(5,8,12,9)(6,14,18,31)(7,20,23,26)(10,25,11,27)(13,29,30,36)(15,16,24,17)(19,32,33,34)(21,28,35,22);(5,9)(6,15)(7,21)(8,12)(10,25)(11,27)(13,29)(14,17)(16,31)(18,24)(19,34)(20,22)(23,35)(26,28)(30,36)(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, 15884);