Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of C7⋊S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C22×C7⋊S4 |
| Fitting, socle | C24×C7 ⋊ D3 |
| derived | C7×A4 ⋊ C23 |
| Sylow | C7 ⋊ C22×S4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | C7⋊S4 |
| commutator, cosocle, hypocenter | C7×A4 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C24×C7 | D3 |
| LCS | G ▷ C7×A4 | |
| UCS | C1 ◁ C22 | |
262 conjugacy classes of subgroups.
| transitive degree | μt(G) = 84 |
68 conjugacy classes — view the full character table.
| maximal subgroup of | #13057101, #13057181, C2×(C7⋊S4)⋊C4, … |
| maximal normal subgroup of | #13057101, #13057181, C2×(C7⋊S4)⋊C4, … |
| maximal quotient of | C2×#10621496, C2×C4×C7⋊S4, C2×C4⋊(C7⋊S4), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1263);