Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C53×M64 ⋊ C2 |
| Sylow | C53 ⋊ #1400 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | #11719921 |
| commutator | C22×C53 | C2×C16 |
| Frattini | C2×C16 | C2×D53 |
| Fitting | C53×M64 | C2 |
| socle | C2×C53 | C22⋊C16 |
| hypercenter | M64 | D53 |
| hypocenter | C53 | #1400 |
| LCS | G ▷ C22×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C8 ◁ C2×C16 ◁ M64 | |
74 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=d,d^2=e,e^2=f,f^2=g,b^a=bc,c^a=cg,d^b=dg,h^b=h^521084 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D53×#1400, C2×#20272854, #27304294, … |
| maximal normal subgroup of | D53×#1400, C2×#20272854, #27304294, … |
| maximal quotient of | #27271430, #27271490, #27271532, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 2649);