Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C26 |
| direct product | C79×#22462 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | D42 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C2×C79 | D4×C22⋊C4 |
| prime radical | C79×#3584 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ G | |
1320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,7,5)(3,12,16,25)(4,17)(6,11,23,10)(8,15,22,14)(9,19)(13,24)(18,20)(21,26)(27,30)(28,31)(29,32)(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)(4,13)(5,22)(6,15)(7,16)(9,24)(10,25)(11,12)(14,23)(17,31)(18,32)(19,30)(20,28)(21,29)(26,27);(1,4)(2,9)(3,13)(5,18)(6,19)(7,21)(8,24)(10,17)(11,26)(12,27)(14,28)(15,30)(16,29)(20,23)(22,32)(25,31);(1,5)(2,7)(3,12)(4,18)(6,10)(8,14)(9,21)(11,23)(13,27)(15,22)(16,25)(17,19)(20,26)(24,28)(29,31)(30,32)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 15450);