Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #67993 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| direct product | C2×#67994 |
| Fitting | C2×#338 ⋊ D3 |
| derived | #338⋊C3 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #4622 |
| commutator, cosocle, hypocenter | #338⋊C3 | C22 |
| Frattini | C22×C4 | C2×S4 |
| Fitting | C2×#338 | D3 |
| hypercenter | C2×C4 | #857 |
| LCS | G ▷ #338⋊C3 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
243 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
36 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11711358, #72213212 |
| maximal normal subgroup of | C2×#11711358, #72213212 |
| maximal quotient of | C2×#72210224 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085737);