Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C26 |
| direct product | C11×#3011 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C11 |
| prime radical | C2×C11×2−1+4 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C2×C11×#41 ◁ G | |
700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 47 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,15)(16,17,21,25)(18,26,29,39)(19,23,30,37)(20,32,34,24)(22,33,36,28)(27,40,41,47)(31,44,43,38)(35,45,46,42);(16,18,21,29)(17,22,25,36)(19,27,30,41)(20,33,34,28)(23,35,37,46)(24,39,32,26)(31,45,43,42)(38,47,44,40);(13,15)(16,19)(17,23)(18,27)(20,31)(21,30)(22,35)(24,38)(25,37)(26,40)(28,42)(29,41)(32,44)(33,45)(34,43)(36,46)(39,47);(19,30)(23,37)(27,41)(31,43)(35,46)(38,44)(40,47)(42,45)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1743);