Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22×C4 |
| direct product | C17×#13241 |
| 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 | C2×C17×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C22×C8×C17 ◁ C2×C4×C8×C17 ◁ G | |
426 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)(19,21)(22,23,28,34,29,36,45,35)(24,31,39,51,40,48,53,52)(25,41,37,30,43,49,38,50)(26,44,46,32,27,47,42,33);(18,19,20,21)(22,24,27,38,45,53,46,43,29,40,26,37,28,39,42,25)(23,30,32,31,34,49,47,51,36,50,33,48,35,41,44,52);(22,25,42,39,28,37,26,40,29,43,46,53,45,38,27,24)(23,31,47,50,35,52,32,49,36,48,44,30,34,51,33,41)1292 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6229);