Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #857 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
| direct product | C2×#67995 |
| derived | C42⋊C3 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #857 |
| commutator, hypocenter | C42⋊C3 | C2×C8 |
| Frattini | C22×C4 | C2×S4 |
| Fitting | C2×C43 | D3 |
| socle | C24 | SL(2,ℤ/4) |
| prime radical | C22×C42⋊C3 | C4 |
| cosocle | C4×C42⋊C3 | C22 |
| LCS | G ▷ C42⋊C3 | |
| UCS | C1 ◁ C2×C4 | |
316 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 316 conjugacy classes of subgroups — too many to draw. Only its 35 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×#11710976, #72214561, #72215399, … |
| maximal normal subgroup of | C2×#11710976, #72214561, #72215399, … |
| maximal quotient of | #72267670, C2×#11710943 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085356);