Finite Group DatabasebetaFinite Group Databasebeta| order | 10688 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | D16×C167 ⋊ C2 |
| Sylow | C167 ⋊ D32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #19128773 |
| commutator, cosocle | C16×C167 | C22 |
| Frattini | C16 | C2×D167 |
| Fitting | D16×C167 | C2 |
| socle | C2×C167 | D16 |
| hypercenter | D16 | D167 |
| hypocenter | C167 | D32 |
| LCS | G ▷ C16×C167 ▷ C8×C167 ▷ C4×C167 ▷ C2×C167 ▷ C167 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ D16 | |
34 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,167|c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,c^b=cd,d^b=de,e^b=ef,g^b=g^166932 conjugacy classes — view the full character table.
| maximal subgroup of | D32×D167, C2×#24923104, #36326445, … |
| maximal normal subgroup of | D32×D167, C2×#24923104, #36326445, … |
| maximal quotient of | #36322315, #36322316, #36322337, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10688, 384);