Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C23 |
| direct product | C17×#13201 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #390 |
| commutator | C22×C4 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C2×C17 | #1955 |
| prime radical | C17×#525 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C8×C17 ◁ G | |
420 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)(18,19,23,29,24,28,43,47,25,30,44,33,22,27,32,38)(20,31,35,49,34,41,46,48,36,39,26,45,42,21,37,40);(18,20,32,46,22,42,44,35,25,36,43,37,24,34,23,26)(19,21,38,49,27,39,33,40,30,41,47,45,28,31,29,48);(18,21,22,39,25,41,24,31)(19,26,28,37,30,35,27,46)(20,33,34,38,36,29,42,47)(23,40,32,45,44,48,43,49)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6189);