Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C42×C8 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C4⋊#857 |
| commutator, cosocle | C4×C42⋊C3 | C22 |
| Frattini | C22×C4 | C2×S4 |
| Fitting | C42×C8 | D3 |
| socle | C23 | C4⋊S4 |
| hypercenter | C8 | #857 |
| hypocenter | C42⋊C3 | D8 |
| LCS | G ▷ C4×C42⋊C3 ▷ C2×C42⋊C3 ▷ C42⋊C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
246 conjugacy classes of subgroups; only the 19 normal subgroups are shown.
This group has 246 conjugacy classes of subgroups — too many to draw. Only its 19 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
Polycyclic presentation:
2^4,3,2^4|b^2=cd,c^2=d,f^2=h,g^2=i,b^a=bc,c^a=cd,e^a=e^2,f^a=fgi,g^a=gi,h^a=hi,f^e=gh,g^e=fg,h^e=i,i^e=hi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72210775, #72226090, #72228326, … |
| maximal normal subgroup of | #72226090, #72228326, #72228328 |
| maximal quotient of | #72210756 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085612);