Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C23 |
| direct product | C11×#2045 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C23×C11 |
| prime radical | C22×C11×D8 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C22×C4×C11 ◁ G | |
296 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,15)(16,17)(18,25)(19,21)(20,24)(22,26)(23,28)(27,30)(29,31);(17,21)(18,26)(19,28)(22,30)(23,24)(25,29);(13,15)(16,18,19,27,20,26,28,31)(17,22,23,30,24,25,21,29)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 777);