Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C4 |
| direct product | C17×#13113 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C17 | C2×D8 |
| commutator | C2×C8 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C22×C17 | C4×D8 |
| prime radical | C23×C17 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C17 ◁ C22×C8×C17 ◁ C2×C4×C8×C17 ◁ G | |
262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 57 |
| transitive degree | μt(G) = 1088 |
A minimal faithful permutation representation — 3 generators on 57 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19)(20,23)(21,22)(24,25)(26,27,32,37,30,35,46,38)(28,36,41,50,40,33,31,47)(29,42,44,54,43,57,45,49)(34,51,53,56,52,48,55,39);(19,22)(23,25)(26,28,32,41,30,40,46,31)(27,33,37,47,35,36,38,50)(29,39,44,51,43,56,45,48)(34,49,53,42,52,54,55,57);(18,20,21,24)(19,23,22,25)(26,29,30,43)(27,34,35,52)(28,39,40,56)(31,48,41,51)(32,44,46,45)(33,49,36,54)(37,53,38,55)(42,50,57,47)986 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6101);