Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#2987 |
| 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 | |
314 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 314 conjugacy classes of subgroups — too many to draw. Only its 248 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,20)(15,18)(16,19)(17,22)(21,26)(23,25)(24,27)(28,29,31,35,32,34,38,41)(30,33,37,40,36,39,42,43);(12,14,16,22)(13,17,19,20)(15,23,24,21)(18,26,27,25)(28,30)(29,33)(31,37)(32,36)(34,39)(35,40)(38,42)(41,43);(12,15,16,24)(13,18,19,27)(14,21,22,23)(17,25,20,26)(29,34)(30,36)(35,41)(37,42);(13,19)(17,20)(18,27)(25,26)(30,36)(33,39)(37,42)(40,43)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1719);