Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C11 · Schur multiplier (H2) = C25 |
| direct product | C11×#2957 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | C22×D4 |
| commutator | C4 | C23×C4×C11 |
| Frattini, cosocle | C2×C4 | C24×C11 |
| socle | C2×C11 | C2×C4×D4 |
| prime radical | C2×C11×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C2×C11×M16 ◁ G | |
452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,17,22,18,21,16,30)(14,23,25,19,26,34,31,33)(15,20,29,35,27,36,28,40)(24,37,39,32,38,42,41,43);(12,14)(13,19)(15,24)(16,31)(17,25)(18,26)(20,32)(21,33)(22,34)(23,30)(27,38)(28,41)(29,39)(35,42)(36,43)(37,40);(12,15,17,29,18,27,16,28)(13,20,22,35,21,36,30,40)(14,24,25,39,26,38,31,41)(19,32,34,42,33,43,23,37);(13,21)(15,27)(19,33)(22,30)(23,34)(24,38)(28,29)(39,41)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1689);