Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1366 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#11956 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #333 |
| commutator | C42 | C22×C4×C17 |
| Frattini, cosocle | C4×C8 | C23×C17 |
| socle | C2×C17 | C2×#333 |
| prime radical | C23×C17 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C4×C17 ◁ C2×C4×C8×C17 ◁ G | |
292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 544 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17)(18,19,22,26,35,27,45,43,23,28,37,44,21,25,34,41)(20,29,32,42,30,40,36,48,31,46,47,49,33,24,38,39);(18,20,23,31)(19,24)(21,30,35,33)(22,36,37,38)(25,29)(26,42)(27,46)(28,40)(32,34,47,45)(39,43)(41,48)(44,49);(18,21,23,35)(19,25,28,27)(20,30,31,33)(22,34,37,45)(24,29,40,46)(26,41,44,43)(32,36,47,38)(39,42,48,49)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 4944);