Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C53×H(3,𝔽4) ⋊ C2 |
| Sylow | C53 ⋊ #1655 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #13133186 |
| commutator | C42×C53 | C23 |
| Frattini | C42 | C22×D53 |
| Fitting | C53×H(3,𝔽4) | C2 |
| socle | C22×C53 | #120 |
| hypercenter | H(3,𝔽4) | D53 |
| hypocenter | C53 | #1655 |
| LCS | G ▷ C42×C53 ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ H(3,𝔽4) | |
262 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|d^2=f,e^2=fg,b^a=bd,c^a=ce,d^a=df,e^a=efg,h^a=h^52,d^b=df,e^b=eg,d^c=dg,e^c=efg514 conjugacy classes — view the full character table.
| maximal subgroup of | #27304821, #27304822, D53×#1655, … |
| maximal normal subgroup of | #27304821, #27304822, D53×#1655, … |
| maximal quotient of | #27272790, #27272837, #27272850, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 3799);