Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C25 |
| direct product | C11×#3416 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C24×C11 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C11×#41 ◁ C11×#131 ◁ G | |
360 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 180 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 — 4 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,17,22)(14,19,23,33)(15,18,28,32)(16,29,27,21)(20,36,31,26)(24,30,37,41)(25,40,38,35)(34,43,42,39);(13,18)(15,26)(16,27)(19,30)(20,21)(22,32)(24,39)(25,38)(28,36)(29,31)(33,41)(34,35)(37,43)(40,42);(12,14)(13,19)(15,24)(16,25)(17,23)(18,30)(20,34)(21,35)(22,33)(26,39)(27,38)(28,37)(29,40)(31,42)(32,41)(36,43);(14,23)(19,33)(24,37)(25,38)(30,41)(34,42)(35,40)(39,43)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 2148);