Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C4 |
| direct product | C17×#13115 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #481 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C2×C17 | #1919 |
| prime radical | C23×C17 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ C2×C17×#97 ◁ G | |
252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,24,29,22,28,25,39)(20,30,33,47,32,26,23,37)(21,34,35,41,36,49,40,46)(27,42,43,48,45,38,44,31);(18,20,24,33,22,32,25,23)(19,26,29,37,28,30,39,47)(21,31,35,42,36,48,40,38)(27,41,43,49,45,46,44,34);(18,21,24,35,22,36,25,40)(19,27,39,44,28,45,29,43)(20,31,33,42,32,48,23,38)(26,41,47,34,30,46,37,49)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6103);