Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C11 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#1284 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | #95 |
| commutator | C2×C4 | C42×C11 |
| Frattini, cosocle | C23×C4 | C22×C11 |
| prime radical | C24×C11 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C23×C4×C11 ◁ G | |
226 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,16)(15,18)(17,19)(20,21)(22,26,27,24)(23,25);(13,16,18,17)(20,22)(21,24)(23,27)(25,26)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 16);