Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#13288 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C79 | C2×#102 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| prime radical | C23×C79 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C2×C42×C79 ◁ C2×C4×C8×C79 ◁ G | |
354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 119 |
| transitive degree | μt(G) = 5056 |
A minimal faithful permutation representation — 3 generators on 119 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,22,27,36)(12,29)(13,20)(14,19)(15,21)(16,25,34,24)(17,23)(18,30)(26,38,28,37)(31,32)(33,39,35,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(9,11,23,39,13,26,31,25,15,27,30,40,14,28,12,24)(10,16,32,38,19,33,17,22,21,34,29,37,20,35,18,36);(1,3,4,8)(2,5,6,7)(9,12,14,30,15,31,13,23)(10,17,20,32,21,18,19,29)(11,24,28,40,27,25,26,39)(16,22,35,38,34,36,33,37)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6276);