Finite Group DatabasebetaFinite Group Databasebeta| order | 8768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C137 ⋊ #364 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22⋊D137 |
| commutator | C4×C137 | C2×C8 |
| Frattini | C2×C8 | C2×D137 |
| Fitting | M32×C137 | C2 |
| socle | C2×C137 | C4⋊C8 |
| hypercenter | M32 | D137 |
| hypocenter | C137 | #364 |
| LCS | G ▷ C4×C137 ▷ C2×C137 ▷ C137 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ M32 | |
46 conjugacy classes of subgroups.
Polycyclic presentation:
2^6,137|a^2=d,b^2=c,c^2=f,d^2=e,e^2=f,b^a=bc,c^a=cf,g^b=g^1361388 conjugacy classes. The character table is too large to display.
| maximal subgroup of | #364×D137, C2×#22577223, #31782553, … |
| maximal normal subgroup of | #364×D137, C2×#22577223, #31782553, … |
| maximal quotient of | #31778852, #31778858, #31778870, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(8768, 369);