Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C23×C4 |
| direct product | C17×#13043 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C17 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4×C17 |
| Frattini, cosocle | C22×D4 | C23×C17 |
| prime radical | C2×C17×B(4,𝔽2) | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C17 ◁ C23×C17 ◁ C22×D4×C17 ◁ G | |
844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(19,21)(22,23)(24,27)(25,32,34,28)(26,29)(30,37)(31,35,33,36);(18,19,20,21)(22,24)(23,27)(25,33)(26,30)(28,36)(29,37)(31,34)(32,35);(23,28)(24,30)(27,35)(29,32)(31,33)(36,37)578 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6031);