Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#1348 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #102 |
| commutator | C2×C8 | C2×C4×C11 |
| Frattini, cosocle | C22×C8 | C22×C11 |
| prime radical | C23×C11 | D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C22×C4×C11 ◁ C22×C8×C11 ◁ G | |
116 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 352 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,18,19,16)(15,17)(21,25)(22,28)(23,30)(26,35)(27,34)(29,32)(31,33);(12,14)(13,16)(15,19)(17,18)(20,21,24,27)(22,29,31,35)(23,32,30,26)(25,33,34,28)319 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 80);