Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of A4×D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×A4×Dic7 |
| derived | C22×C7 ⋊ C2×C3×C4 |
| Hall | C23×Dic7 ⋊ C3 |
| Sylow | C7 ⋊ C2×C4×A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | A4×D7 |
| commutator, hypocenter | C22×C7 | C2×C3×C4 |
| Frattini | C2 | C2×A4×D7 |
| Fitting, socle | C24×C7 | C2×C3 |
| prime radical | C22×C7×A4 | C2 |
| cosocle | C23×C7 | C22×C3 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C22 | |
132 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
80 conjugacy classes — view the full character table.
| maximal subgroup of | Dic7×SL(2,ℤ/4), #13057097, SL(2,ℤ/4)⋊Dic7, … |
| maximal normal subgroup of | Dic7×SL(2,ℤ/4), #13057097, SL(2,ℤ/4)⋊Dic7, … |
| maximal quotient of | C2×A4×C7⋊C8, A4×#1136, C4×A4×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1113);