Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| derived | #997 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×#1020 |
| commutator, hypocenter | #997 | C2×C4 |
| Fitting | C2×#585 | D3 |
| socle | C2 | C2×#5936 |
| prime radical | C2×#5950 | C2 |
| cosocle | #5950 | C22 |
| hypercenter | C2×C4 | #1020 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
899 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 899 conjugacy classes of subgroups — too many to draw. Only its 34 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=c,c^2=i,g^2=i,h^2=i,b^a=bi,d^a=d^2,e^a=ef,g^a=gh,h^a=hi,e^d=f,f^d=ef,g^d=h,h^d=ghi,h^e=hi,g^f=gi,h^g=hi46 conjugacy classes — view the full character table.
| maximal subgroup of | #72215648, #72215726, #72215741, … |
| maximal normal subgroup of | #72215648, #72215726, #72215741, … |
| maximal quotient of | #72227321, #72231922, #72268130, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090048);