Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#13173 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #466 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C2×C17 | #1942 |
| prime radical | C23×C17 | C2×Q16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ C17×#405 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,22,30,25,31,24,28)(20,32,36,49,37,54,34,26)(21,38,40,70,43,71,42,69)(23,45,44,56,46,55,39,29)(27,50,52,73,53,72,51,41)(33,59,63,78,64,79,61,68)(35,65,66,77,67,48,60,58)(47,62,57,75,74,81,76,80);(18,20,22,36,25,37,24,34)(19,26,30,32,31,49,28,54)(21,33,40,63,43,64,42,61)(23,35,44,66,46,67,39,60)(27,47,52,57,53,74,51,76)(29,48,45,58,56,65,55,77)(38,68,70,59,71,78,69,79)(41,62,50,75,73,81,72,80);(18,21,39,72,24,42,46,73,25,43,44,50,22,40,23,41)(19,27,45,70,30,52,56,71,31,53,55,69,28,51,29,38)(20,33,60,80,34,61,67,81,37,64,66,75,36,63,35,62)(26,47,58,59,32,57,65,78,49,74,77,79,54,76,48,68)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6161);