Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #4626⋊C2 |
| commutator, cosocle | #392⋊C3 | C22 |
| Frattini | C22×C4 | C2×S4 |
| Fitting | #2675 | D3 |
| socle | C23 | GL(2,ℤ/4) |
| prime radical | #2675⋊C3 | C2 |
| hypocenter | #95⋊C3 | D4 |
| LCS | G ▷ #392⋊C3 ▷ #95⋊C3 | |
| UCS | C1 ◁ C2 | |
195 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 195 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^3,3,2^5|a^2=gh,c^2=gh,e^2=g,f^2=ghi,b^a=bc,c^a=cgh,d^a=d^2,e^a=efi,f^a=fghi,g^a=gi,h^a=hi,c^b=cgh,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,e^d=fh,f^d=efgh,g^d=ghi,h^d=i,i^d=hi,f^e=fgh24 conjugacy classes — view the full character table.
| maximal subgroup of | #72231937, #72232120, #72340817, … |
| maximal normal subgroup of | #72232120, #72340817, #72340824 |
| maximal quotient of | #72226632, #72238103 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085810);