Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C22×C4 |
| direct product | C11×#1812 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #117 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| socle | C2×C11 | #380 |
| prime radical | C2×C11×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C2×C11×M16 ◁ G | |
260 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 352 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,22)(15,26,28,19)(16,29)(17,21)(18,24)(20,30)(23,36,38,34)(25,35)(27,33,42,41)(31,37,40,43)(32,39);(12,14,16,24,17,25,30,39)(13,18,20,22,21,32,29,35)(15,23,27,37,28,38,42,43)(19,31,33,34,26,40,41,36);(12,15)(13,19)(14,23)(16,27)(17,28)(18,31)(20,33)(21,26)(22,34)(24,37)(25,38)(29,41)(30,42)(32,40)(35,36)(39,43)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 544);