Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#3395 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C11 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C11 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C4×Q8×C11 ◁ G | |
348 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 192 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) = 1408 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,17)(18,19)(20,21,24,27)(22,28,30,35)(23,31,33,26)(25,32,34,29);(12,13,16,17)(14,18,15,19);(12,14,16,15)(13,18,17,19)(20,22,24,30)(21,25,27,34)(23,32,33,29)(26,35,31,28);(12,15,16,14)(13,18,17,19)385 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 2127);