Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#3371 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C11 |
| prime radical | C11×#127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C4×Q8×C11 ◁ G | |
344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,16)(17,18)(20,21)(22,29)(23,27)(24,32)(25,28)(26,31)(30,35)(33,34);(12,13,15,16)(14,18,19,17)(22,30)(26,34)(29,35)(31,33);(20,22,25,30)(21,26,28,34)(23,31,24,33)(27,35,32,29);(12,14,15,19)(13,17,16,18)(20,23,25,24)(21,27,28,32)(22,31,30,33)(26,35,34,29)319 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 2103);