Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C53×#443 ⋊ C2 |
| Sylow | C53 ⋊ #3293 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D212 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#443 | C2 |
| socle | C2×C53 | D42 |
| hypercenter | #443 | D53 |
| hypocenter | C53 | #3293 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #443 | |
444 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 444 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.
Polycyclic presentation:
2^7,53|a^2=g,b^2=g,c^2=g,e^2=g,f^2=g,b^a=be,c^a=cf,e^a=eg,f^a=fg,h^a=h^52,c^b=cg,d^b=de,e^b=eg,f^c=fg,h^c=h^52,e^d=eg757 conjugacy classes — view the full character table.
| maximal subgroup of | #27297489, #27297492, #27297499, … |
| maximal normal subgroup of | #27297489, #27297492, #27297499, … |
| maximal quotient of | #27375406, #27375407, #27375498, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 16315);