Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C11 · Schur multiplier (H2) = C24 |
| direct product | C11×#1870 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C11 | C23 |
| commutator | C23 | C22×C4×C11 |
| Frattini, cosocle | C22×C4 | C23×C11 |
| socle | C23×C11 | C22×C4 |
| prime radical | C22×D4×C11 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C11 ◁ G | |
408 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 136 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,4,5,6,7,8,9,10,11)(13,15)(16,17)(18,19)(20,21,24,26,23,22,25,27);(12,13,14,15)(20,22,24,27,23,21,25,26);(17,19)(21,22)(26,27)484 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 602);