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×#2916 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | C22×D4 |
| commutator | C4 | C23×C4×C11 |
| Frattini, cosocle | C2×C4 | C24×C11 |
| socle | C2×C11 | C22×C4⋊C4 |
| prime radical | C2×C11×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C2×C11×#41 ◁ G | |
440 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)(12,13,17,21,18,22,16,31)(14,23,26,33,27,37,25,19)(15,20,30,35,28,34,29,41)(24,36,40,42,38,43,39,32);(12,14,16,25,18,27,17,26)(13,19,21,23,22,33,31,37)(15,24,29,39,28,38,30,40)(20,32,35,36,34,42,41,43);(12,15)(13,20)(14,24)(16,29)(17,30)(18,28)(19,32)(21,35)(22,34)(23,36)(25,39)(26,40)(27,38)(31,41)(33,42)(37,43);(15,28)(20,34)(24,38)(29,30)(32,42)(35,41)(36,43)(39,40)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1648);