Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#3230 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C11 |
| prime radical | C23×C11 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C2×C42×C11 ◁ G | |
360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,18)(14,19,16,17)(20,21)(22,28)(23,26)(24,27)(25,30)(29,34)(31,33)(32,35);(12,14,15,16)(13,17,18,19)(20,22,24,30)(21,25,27,28)(23,31,32,29)(26,34,35,33);(13,17)(14,16)(18,19)(20,23,24,32)(21,26,27,35)(22,29,30,31)(25,33,28,34);(13,18)(17,19)(21,27)(25,28)(26,35)(33,34)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1962);