Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #4625⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #4625⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4625⋊C2 |
| commutator | #120⋊C3 | C2×C4 |
| Frattini | C24 | C2×S4 |
| Fitting | #2375 | D3 |
| socle | C23 | GL(2,ℤ/4) |
| prime radical, cosocle | C2×#120⋊C3 | C22 |
| hypocenter | C22⋊A4 | C4⋊C4 |
| LCS | G ▷ #120⋊C3 ▷ C22⋊A4 | |
| UCS | C1 ◁ C2 | |
240 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 240 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=d,b^2=c,b^a=bd,e^a=e^2,f^a=fgi,g^a=gh,h^a=hi,f^b=fi,g^b=ghi,f^d=fh,g^d=gi,f^e=g,g^e=fg,h^e=i,i^e=hi32 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#4625⋊C2, #72242975, #72243737, … |
| maximal normal subgroup of | C4×#4625⋊C2 |
| maximal quotient of | #72243737, #72253646 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085937);