Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#1340 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #327 |
| commutator | C2×C8 | C2×C4×C11 |
| Frattini, cosocle | C2×M16 | C22×C11 |
| prime radical | C23×C11 | D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C2×C4×C11 ◁ C2×C11×M16 ◁ G | |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,23,25,40)(15,28,30,39)(16,21)(17,33)(18,22)(19,24)(20,31,37,32)(26,36,27,43)(29,38)(34,41)(35,42);(12,14,18,25)(13,19,22,35)(15,29,30,41)(16,27,17,26)(20,36,37,43)(21,38,33,34)(23,39,40,28)(24,31,42,32)220 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 72);