Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1406 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C11 · Schur multiplier (H2) = C4 |
| direct product | C11×#1406 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C11 | #353 |
| commutator | C42 | C2×C4×C11 |
| Frattini, cosocle | #124 | C22×C11 |
| prime radical | C11×#496 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C11 ◁ C22×C11 ◁ C2×C4×C11 ◁ C11×#124 ◁ G | |
108 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)(12,13,14,19)(15,18,21,26)(16,20,22,27)(17,23,25,24);(13,17)(14,21)(15,24)(18,20)(22,23)(25,26)154 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 138);