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) = C24 |
| direct product | C11×#2984 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C24 |
| commutator | C22 | C23×C4×C11 |
| Frattini, cosocle | C2×C4 | C24×C11 |
| socle | C22×C11 | C23×C4 |
| prime radical | C23×C11 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ G | |
364 conjugacy classes of subgroups; only the 272 normal subgroups are shown.
This group has 364 conjugacy classes of subgroups — too many to draw. Only its 272 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| 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)(12,13,16,17,15,19,14,18)(20,21)(22,28)(23,26)(24,27)(25,30)(29,34)(31,33)(32,35);(12,14,15,16)(13,17,19,18)(20,22,24,30)(21,25,27,28)(23,31,32,29)(26,34,35,33);(12,14,15,16)(13,18,19,17)(20,23,24,32)(21,26,27,35)(22,29,30,31)(25,33,28,34);(13,19)(17,18)(21,27)(25,28)(26,35)(33,34)550 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1716);