Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7396 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#7396 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #353 |
| commutator | C2×M16 | C2×C4×C17 |
| Frattini, cosocle | #407 | C22×C17 |
| socle | C2×C17 | #1343 |
| prime radical | C17×#2010 | C2 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C2×C4×C17 ◁ C22×C4×C17 ◁ C17×#407 ◁ G | |
296 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 42 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)(19,23)(20,26,29,30)(24,27)(25,33)(31,32);(18,19)(20,27)(21,31)(22,25)(23,30)(24,28)(26,33)(29,32)527 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 384);