Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#33637 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C2×C79 | C2×#476 |
| prime radical | C2×#135×C79 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C4×C79 ◁ C2×C4×C8×C79 ◁ G | |
524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,6,11)(3,12,15,24)(4,16)(5,9,21,10)(7,22,25,14)(8,26)(13,28)(17,32)(18,29)(19,23)(20,30)(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);(1,3,5,14,6,15,21,22)(2,7,10,12,11,25,9,24)(4,13,17,26,20,31,19,18)(8,23,28,30,29,32,27,16);(1,4)(2,8)(3,13)(5,17)(6,20)(7,23)(9,27)(10,28)(11,29)(12,30)(14,26)(15,31)(16,24)(18,22)(19,21)(25,32);(1,5,6,21)(2,9,11,10)(3,14,15,22)(4,17,20,19)(7,24,25,12)(8,27,29,28)(13,26,31,18)(16,32,30,23)3634 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26625);