Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×#41×C53 ⋊ C4 |
| Sylow | C53 ⋊ #3428 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C23×#6157 |
| commutator, socle | C2×C53 | C24×C4 |
| Frattini | C2 | C24×#6157 |
| Fitting | C2×#41×C53 | C4 |
| hypercenter | C2×#41 | #6157 |
| hypocenter | C53 | #3428 |
| LCS | G ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C4 ◁ C2×#41 | |
1540 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=g,e^2=f,b^a=bf,d^a=df,h^a=h^30,c^b=cf,h^g=h^52328 conjugacy classes — view the full character table.
| maximal subgroup of | #28306525, #28306528, #28306539, … |
| maximal normal subgroup of | #28306525, #28306528, #28306539, … |
| maximal quotient of | #28328628, #28328631, C4×#16811690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 21490);