Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | C13×#391 ⋊ C2 |
| Sylow | C13 ⋊ #2808 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D13 |
| commutator, socle, cosocle | C23×C13 | C24 |
| Fitting | C13×#391 | C2 |
| prime radical | C22×D52 | C22 |
| hypercenter | #391 | D13 |
| hypocenter | C13 | #2808 |
| LCS | G ▷ C23×C13 ▷ C13 | |
| UCS | C1 ◁ C23 ◁ #391 | |
428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 832 |
Polycyclic presentation:
2^7,13|a^2=eg,b^2=f,c^2=fg,d^2=ef,b^a=be,c^a=cf,d^a=def,h^a=h^12,c^b=cg,d^b=de,h^b=h^12200 conjugacy classes — view the full character table.
| maximal subgroup of | #15751682, #15751686, #15751851, … |
| maximal normal subgroup of | #15751682, #15751686, #15751851, … |
| maximal quotient of | #15791382, #15791393, #15791684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 12492);