Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle | C22 | C2×#1020 |
| commutator, hypocenter | #997 | C2×C4 |
| Fitting | C22×H(5,𝔽2) | D3 |
| prime radical | C22×#997 | C2 |
| cosocle | C2×#997 | C22 |
| hypercenter | C23 | #1020 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C22 ◁ C23 | |
1000 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 36 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:
2^3,3,2^5|a^2=c,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 | #72222024, #72226069, #72226121, … |
| maximal normal subgroup of | #72226121, #72227857, #72228044, … |
| maximal quotient of | #72222024, #72271673, #72340969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090045);