Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C17 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×C17×#324 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C17 | C22⋊C4 |
| commutator | C22 | C2×C4×C8×C17 |
| Frattini, cosocle | C23×C4 | C23×C17 |
| socle | C23×C17 | C2×C22⋊C4 |
| prime radical | C24×C17 | C22×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C17 ◁ C22×C42×C17 ◁ G | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37);(5,6)(7,11,12,9)(8,10)(13,14,15,17,16,18,19,20);(5,7,8,12)(6,9,10,11)1496 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 1139);