Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C2 |
| direct product | C11×#2165 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C11 | #117 |
| commutator | C2×C4 | C22×C4×C11 |
| Frattini, cosocle | C2×C8 | C23×C11 |
| socle | C2×C11 | #432 |
| prime radical | C23×C11 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C11 ◁ C11×#97 ◁ G | |
128 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)(12,13,14,19,15,20,22,27,16,17,21,24,23,18,25,26);(13,17)(14,21)(18,20)(22,25);(13,18,17,20)(14,21)(19,27,24,26)(22,25)352 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 897);