Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #1020 |
| automorphism | ? |
| homology | Abelianization (H1) = C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| derived | C22⋊A4 ⋊ C16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | #1020 |
| commutator, hypocenter | C22⋊A4 | C16 |
| Fitting | C24×C8 | D3 |
| socle | C25 | C3⋊C8 |
| prime radical | C2×C22⋊A4 | C8 |
| cosocle | C8×C22⋊A4 | C2 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C8 | |
308 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 308 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^4,3,2^4|a^2=b,b^2=c,c^2=d,e^a=e^2,f^a=fg,h^a=hi,f^e=g,g^e=fg,h^e=i,i^e=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72253096, #72253098, #72254555, … |
| maximal normal subgroup of | #72253096, C16×#1020 |
| maximal quotient of | #72254549, #72254555 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088644);