Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | #2867 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#857⋊C2 |
| commutator, cosocle | C2×C42⋊C3 | C23 |
| Frattini, socle | C23 | C22×S4 |
| Fitting | #2867 | D3 |
| hypercenter | C4 | #857⋊C2 |
| hypocenter | C42⋊C3 | C2×D4 |
| LCS | G ▷ C2×C42⋊C3 ▷ C42⋊C3 | |
| UCS | C1 ◁ C2 ◁ C4 | |
810 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 810 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.
| transitive degree | μt(G) = 12 |
Polycyclic presentation:
2^4,3,2^4|f^2=h,g^2=i,e^a=e^2,f^a=fg,g^a=gi,h^a=hi,c^b=cd,f^b=fh,g^b=gi,f^c=fh,g^c=gi,f^e=ghi,g^e=fgh,h^e=i,i^e=hi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72209910, #72214473, C2×#11713628, … |
| maximal normal subgroup of | #72214473, C2×#11713628, #72226052, … |
| maximal quotient of | #72219735, C2×#11713628, #72225936, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1088009);