Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C2×C4 |
| direct product | C79×#7732 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C79 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C79 |
| Frattini, cosocle | C2×C42 | C23×C79 |
| prime radical | C22×D4×C79 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C43×C79 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2)(3,5)(4,7)(6,8)(9,10,12,16,13,15,14,11)(17,18)(19,25,27,22)(20,28)(21,24)(23,29)(26,30,32,31)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(2,5)(3,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32);(10,15)(11,16)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 720);