Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C11 · Schur multiplier (H2) = C24×C4 |
| direct product | C11×#3156 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C11 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C11 |
| prime radical | C2×C11×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C11 ◁ C11×#117 ◁ G | |
532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 47 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,15)(16,17,21,25)(18,26,30,40)(19,23,31,37)(20,33,27,24)(22,34,36,29)(28,39,42,47)(32,44,41,38)(35,45,46,43);(16,18,27,34,21,30,20,29)(17,22,33,40,25,36,24,26)(19,28,41,45,31,42,32,43)(23,35,44,47,37,46,38,39);(12,13,14,15)(16,19)(17,23)(18,28)(20,32)(21,31)(22,35)(24,38)(25,37)(26,39)(27,41)(29,43)(30,42)(33,44)(34,45)(36,46)(40,47);(19,31)(23,37)(28,42)(32,41)(35,46)(38,44)(39,47)(43,45)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1888);