Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | M32×#261 |
| Fitting | C13×M32 ⋊ C4 |
| Sylow | C13 ⋊ C4×M32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C22×#261 |
| commutator, socle | C2×C13 | C2×C4×C8 |
| Fitting | C13×M32 | C4 |
| prime radical | C22×D13 | C2×C8 |
| cosocle | C8×D13 | C23 |
| hypercenter | M32 | #261 |
| hypocenter | C13 | C4×M32 |
| LCS | G ▷ C2×C13 ▷ C13 | |
| UCS | C1 ◁ C8 ◁ M32 | |
196 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 208 |
140 conjugacy classes — view the full character table.
| maximal subgroup of | #261×#347, #16789028, #261×#349, … |
| maximal normal subgroup of | #261×#347, #16789028, #261×#349, … |
| maximal quotient of | M32×#1057, #16786579, #16786601, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 20713);