Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #5934 |
| commutator, prime radical, cosocle, hypocenter | #73291 | C2 |
| Frattini, hypercenter | C23 | #1020 |
| Fitting | #2811 | D3 |
| LCS | G ▷ #73291 | |
| UCS | C1 ◁ C22 ◁ C23 | |
180 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 15 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
2,3,2^7|a^2=i,c^2=i,d^2=i,e^2=i,f^2=i,b^a=b^2,c^a=cd,d^a=di,e^a=ef,f^a=fi,g^a=gh,c^b=cdi,d^b=c,e^b=efi,f^b=e,d^c=di,e^c=eh,f^c=fgh,e^d=eg,f^d=fh,f^e=fi28 conjugacy classes — view the full character table.
| maximal subgroup of | #72227829, #72227858, #72227859, … |
| maximal normal subgroup of | #72227858, #72227859, #72227894, … |
| maximal quotient of | #72266674 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088640);