Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | C2×C4×Q8×C53 ⋊ C2 |
| Sylow | C53 ⋊ #3461 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D53 |
| commutator, socle | C22×C53 | C25 |
| Frattini | C22 | C24×D53 |
| Fitting | C2×C4×Q8×C53 | C2 |
| hypercenter | C2×C4×Q8 | D53 |
| hypocenter | C53 | #3461 |
| LCS | G ▷ C22×C53 ▷ C53 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×Q8 | |
1012 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|b^2=f,c^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,h^a=h^52,c^b=cf1084 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #27275505, #27275508, #27275888, … |
| maximal normal subgroup of | #27275505, #27275508, #27275888, … |
| maximal quotient of | #27542787, #27542800, #27543056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 17943);