Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #11710973 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C8 |
No aliases for this group.
| derived | C82⋊C3 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C82⋊C3 | C4 |
| Frattini | #127 | S4 |
| Fitting | #1712 | D3 |
| socle | C22 | #4626 |
| prime radical, cosocle | #1712⋊C3 | C2 |
| LCS | G ▷ C82⋊C3 | |
| UCS | C1 | |
61 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^2,3,2^6|a^2=b,d^2=f,e^2=g,f^2=h,g^2=i,c^a=c^2,d^a=def,e^a=efhi,f^a=fgh,g^a=gh,h^a=hi,d^b=dfh,e^b=egi,f^b=fh,g^b=gi,d^c=efgi,e^c=defhi,f^c=ghi,g^c=fgh,h^c=i,i^c=hi15 conjugacy classes — view the full character table.
| maximal subgroup of | #72245060 |
| maximal normal subgroup of | #72245060 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085354);