Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C11 · Schur multiplier (H2) = C2×C4 |
| direct product | C11×#1275 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C42 |
| commutator | C4 | C4×C8×C11 |
| Frattini, cosocle | C2×C42 | C22×C11 |
| socle | C22×C11 | #95 |
| prime radical | C23×C11 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ G | |
152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,14,16,15,17,18,19)(20,21)(22,29,31,35)(23,27)(24,33)(25,28)(26,30,34,32);(20,22,24,32,25,31,23,30)(21,26,27,29,28,34,33,35)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 7);