Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#1878 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C11 | C2×D4 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ C2×C42×C11 ◁ G | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 1408 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,15)(16,17)(18,19)(20,21,24,26,25,29,23,28)(22,30,32,27,33,34,31,35);(12,13,14,15)(21,26)(23,24)(27,30)(28,29)(31,32)(34,35);(17,19)(20,22,25,33)(21,27,29,35)(23,32,24,31)(26,30,28,34)418 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 610);