Finite Group DatabasebetaFinite Group Databasebeta| order | 6784 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C53×C64 ⋊ C2 |
| Sylow | C53 ⋊ SD128 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D1696 |
| commutator | C32×C53 | C22 |
| Frattini | C32 | C2×D53 |
| Fitting | C53×C64 | C2 |
| socle | C2×C53 | D32 |
| hypercenter | C64 | D53 |
| hypocenter | C53 | SD128 |
| LCS | G ▷ C32×C53 ▷ C16×C53 ▷ C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 | |
38 conjugacy classes of subgroups.
Polycyclic presentation:
2^7,53|a^2=g,c^2=de,d^2=ef,e^2=fg,f^2=g,b^a=bc,c^a=cd,d^a=de,e^a=ef,f^a=fg,h^a=h^52,c^b=cd,d^b=de,e^b=ef,f^b=fg,h^b=h^521699 conjugacy classes. The character table is too large to display.
| maximal subgroup of | D53×SD128, C2×#20272931, #27305145, … |
| maximal normal subgroup of | D53×SD128, C2×#20272931, #27305145, … |
| maximal quotient of | #27271971, #27271974, #27271994, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6784, 2726);