Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#3261 |
| 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 ◁ C11×#117 ◁ G | |
322 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 51 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,16)(14,18)(20,21,25,29)(22,26,31,40)(23,33,35,43)(24,36,38,28)(27,37,42,34)(30,44,45,51)(32,48,47,41)(39,49,50,46);(12,13)(14,17)(15,16)(18,19)(20,22,25,31)(21,26,29,40)(23,30,35,45)(24,32,38,47)(27,39,42,50)(28,41,36,48)(33,44,43,51)(34,46,37,49);(13,16)(17,19)(20,23,25,35)(21,27,29,42)(22,30,31,45)(24,37,38,34)(26,39,40,50)(28,43,36,33)(32,49,47,46)(41,51,48,44);(12,14,15,18)(13,17,16,19)(22,31)(26,40)(30,45)(32,47)(39,50)(41,48)(44,51)(46,49)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1993);