Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | Q32×C53 ⋊ C4 |
| Sylow | C53 ⋊ #1418 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #16811535 |
| commutator | C16×C53 | C2×C4 |
| Frattini | C16 | C2×#6157 |
| Fitting | Q32×C53 | C4 |
| socle | C2×C53 | #357 |
| hypercenter | Q32 | #6157 |
| hypocenter | C53 | #1418 |
| LCS | G ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q32 | |
74 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=d,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,c^b=ceg,d^b=dg,e^b=ef,f^b=fg,h^b=h^52,h^d=h^52169 conjugacy classes — view the full character table.
| maximal subgroup of | D53⋊#1418, C2×#20289869, #28312570, … |
| maximal normal subgroup of | D53⋊#1418, C2×#20289869, #28312570, … |
| maximal quotient of | #28298984, #28298985, #28299064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 19664);