Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C17 · Schur multiplier (H2) = C2 |
| direct product | C17×#7361 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #354 |
| commutator | C42 | C2×C8×C17 |
| Frattini, cosocle | #423 | C22×C17 |
| socle | C2×C17 | #1319 |
| prime radical | C23×C17 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C2×C4×C17 ◁ C22×C4×C17 ◁ C17×#423 ◁ G | |
184 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 46 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)(18,19,21,25,23,27,30,32,24,28,31,33,22,26,20,29);(18,20)(19,25,28,33)(21,23)(22,31)(24,30)(26,29,27,32)578 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 349);