Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of A4×D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C2 |
| direct product | A4×C7⋊C8 |
| derived | C22×C7 ⋊ C3×C8 |
| Hall | C22×C7⋊C8 ⋊ C3 |
| Sylow | C7 ⋊ C8×A4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | A4×D7 |
| commutator, hypocenter | C22×C7 | C3×C8 |
| Fitting, cosocle | C22×C4×C7 | C2×C3 |
| socle | C23×C7 | C3×C4 |
| prime radical | C2×C7×A4 | C4 |
| LCS | G ▷ C22×C7 | |
| UCS | C1 ◁ C4 | |
52 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 168 |
80 conjugacy classes — view the full character table.
| maximal subgroup of | S4×C7⋊C8, (C7⋊S4)⋊C8, #13057034, … |
| maximal normal subgroup of | S4×C7⋊C8, (C7⋊S4)⋊C8, #13057034, … |
| maximal quotient of | SL(2,𝔽3)×C7⋊C8, A4×C7⋊C16, #13052159, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 375);