Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#1336 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #102 |
| commutator | C8 | C2×C8×C11 |
| Frattini, cosocle | C4×C8 | C22×C11 |
| socle | C2×C11 | D4⋊C8 |
| prime radical | C11×#135 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C2×C8×C11 ◁ C4×C8×C11 ◁ G | |
96 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)(12,13,16,20,17,21,27,26,18,22,15,24,25,14,23,19);(12,14)(13,17)(15,19)(16,26)(18,21)(20,23)(22,25)(24,27)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 68);