Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C53×#454 ⋊ C2 |
| Sylow | C53 ⋊ #3021 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #120×D53 |
| commutator | C2×C4×C53 | C24 |
| Frattini | C2×C4 | C23×D53 |
| Fitting | C53×#454 | C2 |
| socle | C2×C53 | C2×#120 |
| hypercenter | #454 | D53 |
| hypocenter | C53 | #3021 |
| LCS | G ▷ C2×C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #454 | |
736 conjugacy classes of subgroups; only the 133 normal subgroups are shown.
This group has 736 conjugacy classes of subgroups — too many to draw. Only its 133 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=g,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,e^b=eg,f^b=fg,e^c=eg,h^c=h^52,h^d=h^52445 conjugacy classes — view the full character table.
| maximal subgroup of | #27538289, #27538291, #27538303, … |
| maximal normal subgroup of | #27538289, #27538291, #27538303, … |
| maximal quotient of | #27356484, #27356531, #27356560, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 13880);