Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C11 · Schur multiplier (H2) = C2×C8 |
| direct product | C11×#1450 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8×C11 | C22 |
| commutator | C2 | C2×C4×C8×C11 |
| Frattini, cosocle | C42 | C23×C11 |
| socle | C22×C11 | C2×C42 |
| prime radical | C23×C11 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8×C11 ◁ G | |
212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(20,21,23,26,24,27,30,33)(22,25,29,32,28,31,34,35);(12,13,14,16,15,17,18,19)(20,22)(21,25)(23,29)(24,28)(26,32)(27,31)(30,34)(33,35);(22,28)(25,31)(29,34)(32,35)880 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 182);