Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#13169 |
| 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 | |
286 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)(22,23,27,32,28,34,42,33)(24,29,26,41,36,31,43,45)(25,37,39,50,40,44,52,51)(30,46,47,53,49,35,48,38);(19,21)(22,24,27,26,28,36,42,43)(23,29,32,41,34,31,33,45)(25,35,39,38,40,46,52,53)(30,44,47,51,49,37,48,50);(22,25,36,46,27,39,43,53,28,40,24,35,42,52,26,38)(23,30,41,51,33,48,29,44,34,49,45,50,32,47,31,37)1292 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6157);