Finite Group DatabasebetaFinite Group Databasebeta| order | 378 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C3 |
| direct product | C3×D7⋊C9 |
| unambiguous semidirect product | (C3×D7)⋊C9 |
| derived, Sylow | C7 ⋊ C2×C3×C9 |
| Hall | C3×C7⋊C9 ⋊ C2 |
| Hall | D7 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C2×C3×C9 |
| Frattini | C3 | C3×AGL(1,𝔽7) |
| Fitting, socle | C32×C7 | C2×C3 |
| prime radical | C32×D7 | C3 |
| cosocle | C3×C7 | C2×C32 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C32 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 189 |
A minimal faithful permutation representation — 2 generators on 19 points:
(5,9,8,6,7,10)(11,12,14,13,15,17,16,18,19);(1,2,3)(4,5,8,9,7,10,6)63 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D7⋊C9, C2×C3×D7⋊C9, C9×D7⋊C9, … |
| maximal normal subgroup of | D3×D7⋊C9, C2×C3×D7⋊C9, C9×D7⋊C9, … |
| maximal quotient of | C3×Dic7⋊C9, C2×C3×D7⋊C9, C3×D7⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(378, 10);