Finite Group DatabasebetaFinite Group Databasebeta| order | 6752 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | SD16×C211 ⋊ C2 |
| Sylow | C211 ⋊ AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D211 |
| commutator | C4×C211 | C23 |
| Frattini | C4 | C22×D211 |
| Fitting | SD16×C211 | C2 |
| socle | C2×C211 | C2×D4 |
| hypercenter | SD16 | D211 |
| hypocenter | C211 | AGL(1,ℤ/8) |
| LCS | G ▷ C4×C211 ▷ C2×C211 ▷ C211 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ SD16 | |
68 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,211|d^2=e,b^a=bd,c^a=ce,d^a=de,f^a=f^210,d^b=de,f^c=f^210746 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/8)×D211, C2×#20270004, #27213943, … |
| maximal normal subgroup of | AGL(1,ℤ/8)×D211, C2×#20270004, #27213943, … |
| maximal quotient of | #27213146, #27213149, #27213185, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6752, 173);