Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | #2811 ⋊ D3 |
| derived | #73291 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C23 | #1020 |
| commutator, cosocle, hypocenter | #73291 | C2 |
| Fitting | #2811 | D3 |
| LCS | G ▷ #73291 | |
| UCS | C1 ◁ C23 | |
358 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 358 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
Polycyclic presentation:
2,3,2^7|c^2=i,d^2=i,e^2=h,f^2=h,b^a=b^2,c^a=cd,d^a=di,e^a=ef,f^a=fh,c^b=cdi,d^b=c,e^b=efh,f^b=e,d^c=di,f^c=fg,e^d=eg,f^e=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #72228063, #72228075, #72241402 |
| maximal normal subgroup of | #72228075, #72241402 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088601);