Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#3244 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C11 |
| prime radical | C11×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C11×#117 ◁ G | |
320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,20)(15,18)(16,19)(17,22)(21,26)(23,25)(24,27)(28,29)(30,37)(31,35)(32,34)(33,38)(36,40)(39,41)(42,43);(12,14,16,22)(13,17,19,20)(15,23,24,21)(18,26,27,25)(28,30)(29,33)(31,38)(32,39)(34,42)(35,37)(36,41)(40,43);(12,15,16,24)(13,18,19,27)(14,21,22,23)(17,25,20,26)(29,34)(30,38)(32,40)(33,41)(35,36)(37,42);(13,19)(17,20)(18,27)(25,26)(29,35)(30,38)(34,36)(39,43)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1976);