Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2169 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C11 | C23 |
| commutator | C22 | C22×C8×C11 |
| Frattini, cosocle | C2×C8 | C23×C11 |
| socle | C22×C11 | C22×C8 |
| prime radical | C2×D4×C11 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C11 ◁ G | |
162 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 162 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 704 |
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)(16,17,20,24,21,18,25,27,19,22,28,30,29,26,23,31);(13,15)(17,22)(18,26)(24,30)(27,31);(16,18,20,27,21,22,25,30,19,26,28,31,29,17,23,24)616 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 901);