Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1902 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C79 · Schur multiplier (H2) = C24 |
| direct product | C79×#12986 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | #386 |
| commutator | C2×C4 | C22×C8×C79 |
| Frattini, cosocle | C22×C8 | C23×C79 |
| socle | C2×C79 | #1869 |
| prime radical | C2×#135×C79 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C22×C8×C79 ◁ G | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 3 generators on 111 points:
(1,2,6,11,5,10,14,30,7,12,26,31,24,9,29,15)(3,13,18,32,16,4,20,21,19,22,8,27,17,23,28,25)(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);(1,3,14,20,24,17,6,18,7,19,29,28,5,16,26,8)(2,4,15,32,9,13,31,25,12,23,30,27,10,22,11,21);(1,4)(2,8)(3,15)(5,22)(6,25)(7,23)(9,20)(10,28)(11,16)(12,18)(13,24)(14,32)(17,31)(19,30)(21,26)(27,29)4582 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 5974);