Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C17 · Schur multiplier (H2) = C2×C4 |
| direct product | C17×#7497 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #353 |
| commutator | C42 | C42×C17 |
| Frattini, cosocle | #532 | C22×C17 |
| socle | C2×C17 | #1393 |
| prime radical | C17×#532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C2×C4×C17 ◁ C22×C4×C17 ◁ C17×#532 ◁ G | |
290 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 290 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 272 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,20,25)(21,23,27,31)(22,26,28,33)(24,29,32,30);(19,23,26,24)(20,27,28,29)578 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 485);