Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#22695 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | D42 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C2×C79 | D4×C4⋊C4 |
| prime radical | C2×#135×C79 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ G | |
808 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,6,12,7,13,24,11)(3,14,17,28,18,27,16,8)(4,19,21,9,22,30,32,26)(5,10,15,25,23,31,20,29)(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);(1,3)(2,8)(4,20)(5,21)(6,17)(7,18)(9,29)(10,30)(11,27)(12,14)(13,28)(15,22)(16,24)(19,31)(23,32)(25,26);(1,4,6,21,7,22,24,32)(2,9,11,19,13,26,12,30)(3,15,17,23,18,20,16,5)(8,25,27,10,28,29,14,31);(1,5)(2,10)(3,4)(6,15)(7,23)(8,26)(9,28)(11,29)(12,25)(13,31)(14,19)(16,32)(17,21)(18,22)(20,24)(27,30)4108 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 15683);