Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #1020 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C24 |
| direct product | C8×#1020 |
| Fitting | C24×C8 ⋊ D3 |
| derived | C22⋊A4 ⋊ C2×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | #1020 |
| commutator, hypocenter | C22⋊A4 | C2×C8 |
| Frattini | C4 | C2×#1020 |
| Fitting | C24×C8 | D3 |
| socle | C25 | C4×D3 |
| prime radical | C2×#1020 | C4 |
| cosocle | C4×C22⋊A4 | C22 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C8 | |
632 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 632 conjugacy classes of subgroups — too many to draw. Only its 31 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 | C8×A4⋊S4, M16×#1020, SD16×#1020, … |
| maximal normal subgroup of | M16×#1020, SD16×#1020, C8×#1020⋊C2, … |
| maximal quotient of | C8×#5934, C8×#5935, #1020×#3633, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090050);