Finite Group DatabasebetaFinite Group Databasebeta| order | 10784 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q16×C337 ⋊ C2 |
| Sylow | C337 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×D337 |
| commutator, cosocle | C4×C337 | C23 |
| Frattini | C4 | C22×D337 |
| Fitting | Q16×C337 | C2 |
| socle | C2×C337 | C2×D4 |
| hypercenter | Q16 | D337 |
| hypocenter | C337 | #135 |
| LCS | G ▷ C4×C337 ▷ C2×C337 ▷ C337 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q16 | |
62 conjugacy classes of subgroups.
Polycyclic presentation:
2^5,337|c^2=e,d^2=e,b^a=bd,d^a=de,f^a=f^336,d^b=de,f^b=f^336,f^c=f^3361190 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #36341736, #36341738, #36341742, … |
| maximal normal subgroup of | #36341736, #36341738, #36341742, … |
| maximal quotient of | #36341300, #36341342, #36341387, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10784, 169);