Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#21559 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | C2×#121 |
| commutator | C2×C4 | C23×C4×C79 |
| Frattini, cosocle | C22×C4 | C24×C79 |
| socle | C2×C79 | C4×#121 |
| prime radical | C79×#3067 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C8×C79 ◁ G | |
868 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,6,13,8,14,7,12)(3,15,17,9,18,25,5,11)(4,19,21,10,22,27,23,28)(16,26,30,29,31,24,20,32)(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);(1,3,6,17,8,18,7,5)(2,9,12,15,14,11,13,25)(4,16,21,30,22,31,23,20)(10,24,19,29,28,26,27,32);(1,4)(2,10)(3,16)(5,20)(6,21)(7,23)(8,22)(9,24)(11,26)(12,19)(13,27)(14,28)(15,29)(17,30)(18,31)(25,32);(1,5,6,3,8,17,7,18)(2,11,13,15,14,9,12,25)(4,20,21,16,22,30,23,31)(10,26,27,29,28,24,19,32)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 14547);