Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #1273 ⋊ D3 |
| derived | #1273⋊C3 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | #67992 |
| commutator, cosocle, hypocenter | #1273⋊C3 | C2 |
| Frattini | C2×C42 | S4 |
| Fitting | #1273 | D3 |
| socle | C23 | #857 |
| LCS | G ▷ #1273⋊C3 | |
| UCS | C1 ◁ C2 | |
113 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 113 conjugacy classes of subgroups — too many to draw. Only its 9 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:
2,3,2^7|c^2=eg,d^2=efgi,e^2=h,f^2=h,g^2=i,b^a=b^2,c^a=cd,d^a=defg,e^a=eghi,f^a=fghi,h^a=hi,c^b=cde,d^b=c,e^b=efgi,f^b=ghi,g^b=fgh,h^b=i,i^b=hi,d^c=defh34 conjugacy classes — view the full character table.
| maximal subgroup of | #72210745, C2×#11710948, #72213079, … |
| maximal normal subgroup of | #72210745, C2×#11710948, #72213079, … |
| maximal quotient of | #72210223, #72210224, #72210225, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085329);