Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#20713 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #381 |
| commutator | C23 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C2×C79 | C2×#381 |
| prime radical | C22×#41×C79 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C23×C4×C79 ◁ G | |
1244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,5,9,6,10,19,25)(3,11,14,21,15,7,13,22)(4,16,17,30,18,8,23,24)(12,20,29,32,28,26,27,31)(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);(1,3)(2,7)(4,12)(5,13)(6,15)(8,20)(9,21)(10,11)(14,19)(16,26)(17,27)(18,28)(22,25)(23,29)(24,31)(30,32);(1,4)(2,8,10,16)(3,12)(5,17)(6,18)(7,20,11,26)(9,24,25,30)(13,27)(14,29)(15,28)(19,23)(21,31,22,32);(1,5,6,19)(2,9,10,25)(3,13,15,14)(4,17,18,23)(7,21,11,22)(8,24,16,30)(12,27,28,29)(20,31,26,32)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 13701);