Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C23×D8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | D4×S4 |
| commutator, cosocle | C2×C4×A4 | C23 |
| Frattini | C2×C4 | C22×S4 |
| Fitting | C23×D8 | D3 |
| socle | C24 | D3×D4 |
| hypercenter | C2×D8 | S4 |
| hypocenter | A4 | #466 |
| LCS | G ▷ C2×C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D8 | |
731 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 731 conjugacy classes of subgroups — too many to draw. Only its 57 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:
2^6,3,2^2|d^2=f,b^a=be,c^a=ce,g^a=g^2,h^a=hi,c^b=cd,d^b=df,d^c=df,h^g=i,i^g=hi58 conjugacy classes — view the full character table.
| maximal subgroup of | D8×GL(2,ℤ/4), #72258868, #72258873 |
| maximal normal subgroup of | D8×GL(2,ℤ/4), #72258868, #72258873 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1087655);