Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C79 · Schur multiplier (H2) = C25 |
| direct product | C79×#33463 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C79 | C2×D8 |
| commutator | C8 | C23×C4×C79 |
| Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C2×C79 | C2×C4×D8 |
| prime radical | C79×#3414 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C79 ◁ C22×C8×C79 ◁ C2×C4×C8×C79 ◁ G | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 2528 |
A minimal faithful permutation representation — 4 generators on 111 points:
(1,2,6,13,8,11,7,23)(3,14,17,31,18,25,24,30)(4,10,19,12,5,21,20,29)(9,22,27,32,26,15,28,16)(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)(2,9)(4,15)(5,22)(6,17)(7,24)(8,18)(10,25)(11,26)(12,30)(13,27)(14,21)(16,19)(20,32)(23,28)(29,31);(1,4,6,19,8,5,7,20)(2,10,13,12,11,21,23,29)(3,15,17,16,18,22,24,32)(9,25,27,30,26,14,28,31);(2,11)(9,26)(10,21)(12,29)(13,23)(14,25)(27,28)(30,31)5056 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26451);