Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×SD16⋊C16 |
| unambiguous semidirect product | SD16⋊(C16×C17) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C17 | C2×D4 |
| commutator | C4 | C22×C16×C17 |
| Frattini, cosocle | C4×C8 | C23×C17 |
| socle | C22×C17 | C8×D4 |
| prime radical | C2×D4×C17 | C2×C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C17 ◁ C4×C16×C17 ◁ G | |
232 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 2176 |
A minimal faithful permutation representation — 3 generators on 41 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,22)(21,23)(24,25)(26,27,29,33,30,34,37,40,31,28,32,35,38,36,39,41);(19,22)(20,25)(23,24);(19,23)(22,24)(26,28,29,35,30,36,37,41,31,27,32,33,38,34,39,40)1496 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 5043);