Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C11 · Schur multiplier (H2) = C26 |
| direct product | Q8×C11×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C11 | C24 |
| commutator | C22 | C23×C4×C11 |
| Frattini, cosocle | C23 | C24×C11 |
| prime radical | C24×C11 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C11 ◁ G | |
820 conjugacy classes of subgroups; only the 392 normal subgroups are shown.
This group has 820 conjugacy classes of subgroups — too many to draw. Only its 392 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 — 4 generators on 27 points:
(1,2,3,4)(5,8,7,6)(17,18,19,20,21,22,23,24,25,26,27);(1,5,3,7)(2,6,4,8);(9,10,12,14)(11,13,15,16);(10,13)(14,16)550 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1072);