Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#13286 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | C2×D8 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C22×C17 | C2×#102 |
| prime radical | C23×C17 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C2×C42×C17 ◁ C2×C4×C8×C17 ◁ G | |
342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 3 generators on 57 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,24)(21,22)(23,25)(26,27,32,37,33,38,31,49)(28,39,43,51,44,56,42,54)(29,36,30,48,47,50,46,35)(34,40,52,57,53,45,55,41);(19,22)(23,24)(26,28,30,41,32,43,47,40,33,44,46,57,31,42,29,45)(27,34,50,56,49,55,48,51,38,53,36,39,37,52,35,54);(18,20,21,25)(19,23,22,24)(26,29,31,46,33,47,32,30)(27,35,37,36,38,48,49,50)(28,40,42,41,44,45,43,57)(34,51,52,56,53,54,55,39)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6274);