Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#2228 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C11 |
| prime radical | C2×C11×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C42×C11 ◁ C4×C8×C11 ◁ G | |
136 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 64 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,17)(14,18,19,16)(20,21,22,25)(23,26,29,33)(24,32,30,28)(27,35,34,31);(12,14,15,19)(13,16,17,18)(21,25)(26,33)(27,34)(28,32);(21,26)(23,31)(24,30)(25,33)(27,28)(29,35)(32,34)319 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 960);