Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C4×C13×M32 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8×C13 | C22 |
| commutator | C2 | C2×C4×C8×C13 |
| Frattini, cosocle | C2×C8 | C23×C13 |
| socle | C22×C13 | C22×C8 |
| prime radical | C23×C13 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8×C13 ◁ G | |
196 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 176 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4)(21,22,23,24,25,26,27,28,29,30,31,32,33);(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20);(6,14)(8,16)(10,18)(12,20)1040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 839);