Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #73081 |
| commutator, cosocle | #5950 | C22 |
| Frattini | C22×C4 | C2×S4 |
| Fitting | #3027 | D3 |
| prime radical | #3027⋊C3 | C2 |
| hypercenter | C4 | #1020⋊C2 |
| hypocenter | #997 | D4 |
| LCS | G ▷ #5950 ▷ #997 | |
| UCS | C1 ◁ C2 ◁ C4 | |
269 conjugacy classes of subgroups; only the 15 normal subgroups are shown.
This group has 269 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.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
2^3,3,2^5|a^2=i,c^2=i,e^2=i,f^2=i,b^a=bc,c^a=ci,d^a=d^2,e^a=ef,f^a=fi,g^a=gh,c^b=ci,e^b=egi,f^b=fhi,e^d=f,f^d=efi,g^d=h,h^d=gh,f^e=fi,h^e=hi,g^f=gi25 conjugacy classes — view the full character table.
| maximal subgroup of | #72236743, #72236759, #72236766, … |
| maximal normal subgroup of | #72236743, #72236759, #72236766, … |
| maximal quotient of | C2×#11711646, #72341859, #72341942, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086027);