Finite Group DatabasebetaFinite Group Databasebeta| order | 4352 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C17 · Schur multiplier (H2) = C22 |
| direct product | C17×#13127 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C17 | #388 |
| commutator | C22×C4 | C22×C4×C17 |
| Frattini, cosocle | C22×C8 | C23×C17 |
| socle | C2×C17 | #1924 |
| prime radical | C2×#41×C17 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C17 ◁ C22×C8×C17 ◁ G | |
300 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)(18,19,23,29,22,27,31,26,24,28,40,46,38,47,21,36)(20,30,32,45,34,25,37,43,35,44,39,48,33,42,49,41);(18,20,31,49,38,33,23,39,24,35,21,37,22,34,40,32)(19,25,26,45,47,30,29,41,28,42,36,48,27,44,46,43);(18,21,22,23,24,31,38,40)(19,26,27,46,28,36,47,29)(20,32,33,49,35,39,34,37)(25,41,30,48,42,43,44,45)782 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4352, 6115);