Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #1020⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#1020⋊C4 |
| Fitting | C2×#379 ⋊ D3 |
| derived | C22⋊A4 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #1020⋊C2 |
| commutator, hypocenter | C22⋊A4 | C22×C4 |
| Frattini | C23 | C22×S4 |
| Fitting | C2×#379 | D3 |
| socle | C24 | C2×S4 |
| prime radical | C22×#1020 | C2 |
| cosocle | C2×C22⋊A4 | C23 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C22 | |
1322 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 1322 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 24 |
56 conjugacy classes — view the full character table.
| maximal subgroup of | #72215370, C2×#11713200, #72218429, … |
| maximal normal subgroup of | #72215370, C2×#11713200, #72220578, … |
| maximal quotient of | #72215370, C2×#72215879, D3×#1020⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1087122);