Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C13 ⋊ #2971 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×#261 |
| commutator, socle | C22×C13 | C23×C4 |
| Fitting, prime radical | C2×C13×#41 | C4 |
| cosocle | C2×Dic13 | C24 |
| hypercenter | C2×#41 | #261 |
| hypocenter | C13 | #2971 |
| LCS | G ▷ C22×C13 ▷ C13 | |
| UCS | C1 ◁ C22 ◁ C2×#41 | |
402 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 168 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=f,c^2=f,d^2=e,f^2=g,b^a=be,c^a=cg,d^a=dg,h^a=h^5,c^b=cg,d^c=dg,h^c=h^8,h^d=h^12,h^f=h^12110 conjugacy classes — view the full character table.
| maximal subgroup of | #16783812, #16783814, #16783820, … |
| maximal normal subgroup of | #16783812, #16783814, #16783820, … |
| maximal quotient of | #16797320, #16797393, Q8×#88110, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 21356);