Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C53×SD64 ⋊ C2 |
| Sylow | C53 ⋊ #2263 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D16×D53 |
| commutator | C16×C53 | C23 |
| Frattini | C16 | C22×D53 |
| Fitting | C53×SD64 | C2 |
| socle | C2×C53 | C2×D16 |
| hypercenter | SD64 | D53 |
| hypocenter | C53 | #2263 |
| LCS | G ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ SD64 | |
112 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Polycyclic presentation:
2^7,53|d^2=ef,e^2=fg,f^2=g,b^a=bd,c^a=cg,d^a=de,e^a=ef,f^a=fg,h^a=h^52,d^b=de,e^b=ef,f^b=fg,h^c=h^52520 conjugacy classes — view the full character table.
| maximal subgroup of | D53×#2263, C2×#20277029, #27542235, … |
| maximal normal subgroup of | D53×#2263, C2×#20277029, #27542235, … |
| maximal quotient of | #27304530, #27304533, #27304598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 6824);