Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C22×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C8 |
| Fitting | #3573 | D3 |
| socle | C2 | C22×A4⋊C8 |
| prime radical | #5945 | C4 |
| cosocle | #131⋊C3 | C23 |
| hypercenter | #131 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C8 ◁ #131 | |
419 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
2^5,3,2^3|a^2=d,d^2=e,e^2=i,g^2=i,h^2=i,f^a=f^2,g^a=gh,h^a=hi,c^b=ci,g^f=h,h^f=ghi,h^g=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088508);