Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C23×C4 |
| direct product | C11×#3137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C11 |
| prime radical | C2×C11×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C11×#117 ◁ G | |
372 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)(12,13)(14,17)(15,16)(18,19)(20,21,25,27)(22,28,29,35)(23,24,31,32)(26,33,34,30);(13,16)(17,19)(20,22,24,30,25,29,32,33)(21,26,23,28,27,34,31,35);(12,14,15,18)(13,17,16,19)(20,23,24,27,25,31,32,21)(22,28,30,34,29,35,33,26);(22,29)(26,34)(28,35)(30,33)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1869);