Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C23 |
| direct product | C79×#12820 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | #459 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C2×#135×C79 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ C2×C4×C8×C79 ◁ G | |
414 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)(7,18)(8,23)(9,16)(10,14)(11,17)(12,20)(13,25)(15,21)(19,33)(22,29)(24,28)(26,36)(27,32)(30,35)(31,34)(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);(5,7,9,20,11,22,10,21)(6,12,14,18,17,15,16,29)(8,19,24,34,26,35,27,25)(13,28,30,23,31,32,33,36);(5,8)(6,13)(7,19)(9,24)(10,27)(11,26)(12,28)(14,30)(15,32)(16,33)(17,31)(18,23)(20,34)(21,25)(22,35)(29,36)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5808);