Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#33497 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C79 | C24 |
| commutator | C22 | C23×C8×C79 |
| Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C22×C79 | C23×C8 |
| prime radical | C23×C79 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C79 ◁ G | |
486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,19,23,20,24,25,30,21,22,26,28,27,29,31,32)(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);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,22)(23,28)(24,29)(30,32);(2,8)(6,9)(7,16)(14,15)6952 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26485);