Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #7855 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C79 · Schur multiplier (H2) = C22 |
| direct product | C79×#12657 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C79 | C2×C42 |
| commutator | C2×C4 | C2×C42×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C22×C79 | C2×#97 |
| prime radical | C23×C79 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C79 ◁ G | |
334 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)(7,18,21,31)(8,14)(9,15)(10,27)(11,16)(12,17)(13,20,29,23)(19,34,35,30)(22,36,24,28)(25,32)(26,33)(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)(5,7,10,22,11,23,25,19,12,21,8,24,9,20,26,35)(6,13,14,30,16,31,33,36,17,29,27,34,15,18,32,28);(5,8,9,25,12,10,11,26)(6,14,15,32,17,27,16,33)(7,19,23,24,21,35,20,22)(13,28,31,30,29,36,18,34)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5645);