Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #1020 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×#5936 |
| derived | C22⋊A4 ⋊ C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | #1020 |
| commutator, hypocenter | C22⋊A4 | C42 |
| Frattini | C22 | C2×#1020 |
| Fitting | C25×C4 | D3 |
| socle | C26 | C2×D3 |
| prime radical, cosocle | C22×C22⋊A4 | C22 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C2×C4 | |
1710 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 1710 conjugacy classes of subgroups — too many to draw. Only its 55 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 | C42×#1020, C4×A4⋊SL(2,ℤ/4), C4×#87524, … |
| maximal normal subgroup of | C42×#1020, C4×#87524, C4×#73081, … |
| maximal quotient of | #72227309, #72227321, AGL(1,𝔽5)×#5936, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1089949);