Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7515 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C17 · Schur multiplier (H2) = C2 |
| direct product | C17×#7516 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C17 | #1402 |
| commutator | #127 | C2×C4×C17 |
| Frattini, cosocle | #493 | C22×C17 |
| prime radical | C17×#493 | C22 |
| LCS | G ▷ #127 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C17 ◁ C22×C17 ◁ C2×C4×C17 ◁ C42×C17 ◁ C17×#493 ◁ G | |
162 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 30 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)(20,25,28,33)(21,29,30,32)(22,26)(23,27,24,31);(18,20,22,28)(19,23,26,29)(21,27,31,30)(24,25,32,33)374 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 504);