Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#8248 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C17 |
| Frattini, cosocle | C2×C42 | C23×C17 |
| socle | C22×C17 | C4×C22⋊C4 |
| prime radical | C17×#121 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C2×C4×C8×C17 ◁ G | |
392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1088 |
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)(20,26)(21,29)(22,25)(23,28)(24,30)(27,32)(31,33)(34,35,39,42,40,36,38,44)(37,41,46,48,47,43,45,49);(18,20)(19,23,25,31)(21,28)(22,27)(24,26,29,32)(30,33)(34,36,40,35)(37,41,47,43)(38,42,39,44)(45,49,46,48);(20,27)(23,31)(26,32)(28,33)(34,37,39,46,40,47,38,45)(35,41,42,48,36,43,44,49)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 1236);