Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#1408 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #353 |
| commutator | C42 | C2×C4×C11 |
| Frattini, cosocle | #127 | C22×C11 |
| prime radical | C11×#493 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C11 ◁ C2×C4×C11 ◁ C11×#127 ◁ G | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,21,22,17)(15,19,25,18)(16,20)(23,26,24,27);(12,14)(13,17)(15,24)(16,23)(18,26)(19,21)(20,27)(22,25)187 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 140);