Finite Group DatabasebetaFinite Group Databasebeta| order | 1408 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C11 · Schur multiplier (H2) = C25×C4 |
| direct product | C42×D4×C11 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C11 | C22 |
| commutator | C2 | C22×C42×C11 |
| Frattini, cosocle | C23 | C24×C11 |
| socle | C23×C11 | C24 |
| prime radical | C22×D4×C11 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C11 ◁ G | |
924 conjugacy classes of subgroups; only the 576 normal subgroups are shown.
This group has 924 conjugacy classes of subgroups — too many to draw. Only its 576 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 704 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3,4)(13,14,15,16,17,18,19,20,21,22,23);(5,6,7,8);(9,10,11,12);(10,12)880 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1408, 1003);