Finite Group DatabasebetaFinite Group Databasebeta| order | 8768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | Q32×C137 ⋊ C2 |
| Sylow | C137 ⋊ SD64 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #17984029 |
| commutator | C16×C137 | C22 |
| Frattini | C16 | C2×D137 |
| Fitting | Q32×C137 | C2 |
| socle | C2×C137 | D16 |
| hypercenter | Q32 | D137 |
| hypocenter | C137 | SD64 |
| LCS | G ▷ C16×C137 ▷ C8×C137 ▷ C4×C137 ▷ C2×C137 ▷ C137 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ Q32 | |
32 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,137|a^2=f,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^136767 conjugacy classes — view the full character table.
| maximal subgroup of | SD64×D137, C2×#22577240, #31783096, … |
| maximal normal subgroup of | SD64×D137, C2×#22577240, #31783096, … |
| maximal quotient of | #31778966, #31778970, #31778990, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8768, 386);