Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C11 · Schur multiplier (H2) = C23×C4 |
| direct product | C11×#2347 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C11 | C24 |
| commutator | C22 | C23×C4×C11 |
| Frattini, cosocle | C23 | C24×C11 |
| prime radical | C24×C11 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ G | |
504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,15)(16,17,20,19)(18,22,23,21)(24,25,29,28)(26,31,27,30);(17,19)(21,22)(24,26,29,27)(25,30,28,31);(16,18,20,23)(17,21,19,22)(24,27,29,26)(25,30,28,31);(16,19,20,17)(18,22,23,21)(24,28,29,25)(26,30,27,31)550 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1079);