Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C26 |
| direct product | C2×C11×#452 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C11 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C11 |
| prime radical | C22×C11×#41 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ C23×C4×C11 ◁ G | |
760 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 33 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33);(3,4)(5,6)(7,8,12,15)(9,16,18,21)(10,14,11,19)(13,20,22,17);(7,9)(8,13)(10,17)(11,20)(12,18)(14,21)(15,22)(16,19);(4,6)(7,10,12,11)(8,14,15,19)(9,17,18,20)(13,21,22,16)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1733);