Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C23 |
| direct product | C2×C11×#367 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C11 | D8 |
| commutator | C8 | C22×C4×C11 |
| Frattini, cosocle | C2×C8 | C23×C11 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ C22×C4×C11 ◁ C22×C8×C11 ◁ G | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1408 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33);(3,4,5,6)(8,12)(9,16)(10,17)(13,18)(14,22)(15,21)(19,20);(7,8,13,22,10,14,9,12,11,15,16,20,17,19,18,21)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 889);