Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | #1020 |
| commutator, hypocenter | C22⋊SL(2,𝔽3) | C8 |
| Fitting | C22×#131 | D3 |
| socle | C23 | A4⋊C8 |
| prime radical | #5948 | C4 |
| cosocle | #85665 | C2 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C8 | |
250 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 250 conjugacy classes of subgroups — too many to draw. Only its 21 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^3,3,2^5|a^2=b,b^2=c,c^2=i,g^2=i,h^2=i,d^a=d^2,e^a=ef,g^a=gh,h^a=hi,e^d=f,f^d=ef,g^d=h,h^d=ghi,h^g=hi60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088645);