Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #4621 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×SL(2,ℤ/8) |
| derived | #95⋊C3 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | #857 |
| commutator, hypocenter | #95⋊C3 | C2×C4 |
| Frattini | C24 | C2×S4 |
| Fitting | C22×#95 | D3 |
| socle | C25 | S4 |
| prime radical | C22×#95⋊C3 | C2 |
| cosocle | C2×#95⋊C3 | C22 |
| LCS | G ▷ #95⋊C3 | |
| UCS | C1 ◁ C23 | |
488 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
60 conjugacy classes — view the full character table.
| maximal subgroup of | #72213078, C2×#11711236, #72214627, … |
| maximal normal subgroup of | #72213078, C2×#11711236, #72214627, … |
| maximal quotient of | C2×#72210223, #72211408, C2×#11710958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085358);