Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C79 · Schur multiplier (H2) = C4 |
| direct product | C79×#13115 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #481 |
| commutator | C2×C8 | C22×C4×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | #1919 |
| prime radical | C23×C79 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ C2×#97×C79 ◁ G | |
252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,7,12,5,10,18,13)(3,14,16,26,6,8,24,11)(4,17,20,32,22,25,21,29)(9,23,27,31,28,15,30,19)(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);(1,3,7,16,5,6,18,24)(2,8,12,11,10,14,13,26)(4,15,20,19,22,23,21,31)(9,25,27,29,28,17,30,32);(1,4,18,21,5,22,7,20)(2,9,12,27,10,28,13,30)(3,15,24,31,6,23,16,19)(8,25,11,29,14,17,26,32)3160 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6103);