Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#7054 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | #99 |
| commutator | C23 | C2×C16×C17 |
| Frattini, cosocle | C2×C22⋊C8 | C22×C17 |
| socle | C22×C17 | #323 |
| prime radical | C25×C17 | C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C22×C4×C17 ◁ C2×C17×C22⋊C8 ◁ G | |
476 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 2 generators on 41 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,24,25,22)(21,23)(26,27,28,31,29,32,34,37,30,33,35,38,36,39,40,41);(19,22)(23,24)884 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 42);