Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| isoclinism | isoclinic class of C3⋊#226 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7 · Schur multiplier (H2) = C3 |
| direct product | C7×C3⋊GL(2,𝔽3) |
| unambiguous semidirect product | C3⋊(C7×GL(2,𝔽3)) |
| Fitting | C3×C7×Q8 ⋊ D3 |
| derived | C3×SL(2,𝔽3) ⋊ C2×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | C3⋊S4 |
| commutator, cosocle, hypocenter | C3×SL(2,𝔽3) | C2×C7 |
| Frattini | C2 | C7×C3⋊S4 |
| Fitting | C3×C7×Q8 | D3 |
| socle | C2×C3×C7 | S4 |
| LCS | G ▷ C3×SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C7 | |
92 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
105 conjugacy classes — view the full character table.
| maximal subgroup of | C7×SO+(4,𝔽3), ΓL(1,𝔽8)×C3⋊GL(2,𝔽3), D7×C3⋊GL(2,𝔽3), … |
| maximal normal subgroup of | ΓL(1,𝔽8)×C3⋊GL(2,𝔽3), D7×C3⋊GL(2,𝔽3) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 632);