Finite Group DatabasebetaFinite Group Databasebeta| order | 1176 |
| isoclinism | isoclinic class of D7×AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | D7×ΓL(1,𝔽8)⋊C4 |
| unambiguous semidirect product | (D7×ΓL(1,𝔽8))⋊C4 |
| derived, Sylow | C72 ⋊ C2×C3×C4 |
| Hall | D7×Dic7 ⋊ C3 |
| Hall | C7×ΓL(1,𝔽8) ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D7×AGL(1,𝔽7) |
| commutator, hypocenter | C72 | C2×C3×C4 |
| Fitting, socle, cosocle | C2×C72 | C22×C3 |
| prime radical | C2×D7×ΓL(1,𝔽8) | C2 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C2 | |
67 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
70 conjugacy classes — view the full character table.
| maximal subgroup of | #72245004, #72245008, C4×D7×AGL(1,𝔽7), … |
| maximal normal subgroup of | #72245004, #72245008, C4×D7×AGL(1,𝔽7), … |
| maximal quotient of | #72245756, D7×ΓL(1,𝔽8)⋊AGL(1,𝔽5) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1176, 81);