Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C4 |
| direct product | C11×#1718 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C11 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C11 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C42×C11 ◁ G | |
164 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 164 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 1408 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,16,19)(14,20,21,27)(15,23,24,18)(17,25,26,22)(28,29)(30,36)(31,33)(32,40)(34,38)(35,39)(37,42)(41,43);(12,14,15,22,16,21,24,25)(13,17,18,20,19,26,23,27)(29,33)(34,41)(35,40)(36,42);(14,21)(17,26)(20,27)(22,25)(28,30,32,38,31,37,39,43)(29,34,35,36,33,41,40,42)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 450);