Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C11 · Schur multiplier (H2) = C22 |
| direct product | C11×#2091 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C23×C11 |
| socle | C2×C11 | #399 |
| prime radical | C24×C11 | Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C22×C4×C11 ◁ G | |
222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 176 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13,15,19,16,18,21,25)(14,20,22,26,23,27,24,17);(12,14,21,22,16,23,15,24)(13,17,25,20,18,26,19,27);(13,18)(19,25)286 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 823);