Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×SD32 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D8×S4 |
| commutator, cosocle | C8×A4 | C23 |
| Frattini | C8 | C22×S4 |
| Fitting | C22×SD32 | D3 |
| socle | C23 | D3×D8 |
| hypercenter | SD32 | S4 |
| hypocenter | A4 | #508 |
| LCS | G ▷ C8×A4 ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ SD32 | |
374 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3,2^2|c^2=f,d^2=ef,e^2=f,b^a=bf,c^a=cf,g^a=g^2,h^a=hi,c^b=cd,d^b=de,e^b=ef,d^c=de,e^c=ef,h^g=i,i^g=hi55 conjugacy classes — view the full character table.
| maximal subgroup of | S4×#508 |
| maximal normal subgroup of | S4×#508 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088588);