Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C32×C53 ⋊ C4 |
| Sylow | C53 ⋊ #1426 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16811550 |
| commutator | C16×C53 | C2×C4 |
| Frattini | C16 | C2×#6157 |
| Fitting | C32×C53 | C4 |
| socle | C2×C53 | #367 |
| hypercenter | C32 | #6157 |
| hypocenter | C53 | #1426 |
| LCS | G ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 | |
52 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=d,b^2=cef,c^2=efg,e^2=fg,f^2=g,b^a=bc,c^a=ce,e^a=ef,f^a=fg,h^a=h^30,d^b=dg,h^d=h^52442 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#20289880, D53⋊#1426, #28312605, … |
| maximal normal subgroup of | C2×#20289880, D53⋊#1426, #28312605, … |
| maximal quotient of | #28299153, #28299155, #28299186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 19675);