Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#13170 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | C2×D8 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C22×C17 | C4×D8 |
| prime radical | C23×C17 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C22×C8×C17 ◁ C2×C4×C8×C17 ◁ G | |
278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,21)(22,23,27,32,28,34,43,33)(24,29,26,42,36,31,44,46)(25,37,40,51,41,45,50,52)(30,38,47,53,49,35,48,39);(19,21)(22,24,27,26,28,36,43,44)(23,29,32,42,34,31,33,46)(25,38,40,53,41,35,50,39)(30,37,47,51,49,45,48,52);(22,25,36,38,27,40,44,53,28,41,24,35,43,50,26,39)(23,30,42,52,33,48,29,45,34,49,46,51,32,47,31,37)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6158);