Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C17 · Schur multiplier (H2) = C29×C4 |
| direct product | C22×C17×#410 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C17 | C22⋊C4 |
| commutator | C22 | C24×C4×C17 |
| Frattini, cosocle | C23 | C25×C17 |
| socle | C23×C17 | C2×C22⋊C4 |
| prime radical | C23×#41×C17 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C17 ◁ C24×C4×C17 ◁ G | |
5278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37);(3,4);(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)1496 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 53055);