Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#1478 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C2×D4 |
| commutator | C4 | C22×C8×C13 |
| Frattini, cosocle | C42 | C23×C13 |
| socle | C22×C13 | C2×C22⋊C4 |
| prime radical | C22×D4×C13 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C42×C13 ◁ G | |
312 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 124 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,20)(17,19)(18,21)(22,23,24,26,25,27,28,29);(15,19)(16,17)(20,21);(14,16,18,17)(15,20,21,19)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 210);