Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C11 · Schur multiplier (H2) = C23×C4 |
| direct product | C11×#2921 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C11 | C24 |
| commutator | C22 | C23×C4×C11 |
| Frattini, cosocle | C2×C4 | C24×C11 |
| socle | C22×C11 | C23×C4 |
| prime radical | C24×C11 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C11 ◁ G | |
392 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 264 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 352 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11)(12,13)(14,17)(15,16)(18,19)(20,21,23,27,24,26,30,33)(22,25,29,32,28,31,34,35);(13,16)(17,19)(20,22)(21,25)(23,29)(24,28)(26,31)(27,32)(30,34)(33,35);(12,14,15,18)(13,17,16,19)(22,28)(25,31)(29,34)(32,35);(21,26)(25,31)(27,33)(32,35)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1653);