Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2098 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C23×C11 |
| socle | C2×C11 | #399 |
| prime radical | C2×C11×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ G | |
210 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 80 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)(12,13,16,21,17,22,18,32)(14,23,25,35,27,39,26,19)(15,28,30,38,31,20,33,37)(24,40,29,43,41,36,42,34);(12,14,18,25,17,27,16,26)(13,19,32,23,22,35,21,39)(15,29,33,41,31,42,30,24)(20,36,38,34,28,40,37,43);(12,15)(13,20)(14,24)(16,30)(17,31)(18,33)(19,34)(21,37)(22,28)(23,40)(25,29)(26,42)(27,41)(32,38)(35,43)(36,39)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 830);