Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C26 |
| direct product | D4×#1020 |
| Fitting | C24×D4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×#1020 |
| commutator, cosocle | C2×C22⋊A4 | C23 |
| Fitting | C24×D4 | D3 |
| socle | C25 | C22×D3 |
| hypercenter | D4 | #1020 |
| hypocenter | C22⋊A4 | C2×D4 |
| LCS | G ▷ C2×C22⋊A4 ▷ C22⋊A4 | |
| UCS | C1 ◁ C2 ◁ D4 | |
2712 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 2712 conjugacy classes of subgroups — too many to draw. Only its 49 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
50 conjugacy classes — view the full character table.
| maximal subgroup of | D4×A4⋊S4, S4×#1020, C2×D4×#1020, … |
| maximal normal subgroup of | C2×D4×#1020, #72220492, D4×#1020⋊C2, … |
| maximal quotient of | #72215654, #72215704, #72215710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1090183);