Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #67992 |
| commutator, prime radical, cosocle, hypocenter | #1273⋊C3 | C2 |
| Frattini | C2×C42 | S4 |
| Fitting | #1273 | D3 |
| socle | C23 | #857 |
| LCS | G ▷ #1273⋊C3 | |
| UCS | C1 ◁ C2 | |
84 conjugacy classes of subgroups.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2,3,2^7|a^2=e,c^2=efg,d^2=egi,f^2=h,g^2=i,b^a=b^2,c^a=cd,d^a=deg,f^a=fg,g^a=gi,h^a=hi,c^b=cde,d^b=cghi,f^b=ghi,g^b=fgh,h^b=i,i^b=hi,d^c=de34 conjugacy classes — view the full character table.
| maximal subgroup of | #72210748, C2×#11710949, #72213079, … |
| maximal normal subgroup of | #72210748, C2×#11710949, #72213079, … |
| maximal quotient of | #72210223, C2×#11710949 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085330);