Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #857 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C22×#4623 |
| derived | C42⋊C3 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #857 |
| commutator, hypocenter | C42⋊C3 | C22×C4 |
| Frattini | C23 | C22×S4 |
| Fitting | C23×C42 | D3 |
| socle | C25 | S4 |
| prime radical | C23×C42⋊C3 | C2 |
| cosocle | C2×C42⋊C3 | C23 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C23 | |
1012 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 1012 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
80 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72210223, #72213068, C2×#11711335, … |
| maximal normal subgroup of | #72213068, C2×#11711335, C2×#11711337, … |
| maximal quotient of | #72211388, C2×#11710976, C2×#11710979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1086921);