Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #11710969 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, prime radical, cosocle, hypocenter | #1309⋊C3 | C2 |
| Frattini | C2×C42 | S4 |
| Fitting | #1309 | D3 |
| socle | C22 | #4621 |
| LCS | G ▷ #1309⋊C3 | |
| UCS | C1 | |
61 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2,3,2^7|a^2=e,c^2=efg,d^2=egh,f^2=h,g^2=i,b^a=b^2,c^a=cd,d^a=deg,f^a=fghi,h^a=hi,c^b=cdei,d^b=cg,f^b=ghi,g^b=fgh,h^b=i,i^b=hi,d^c=de,e^c=ehi,f^c=fi,g^c=gh,e^d=ei,f^d=fhi,g^d=gi19 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085350);