Finite Group DatabasebetaFinite Group Databasebeta| order | 378 |
| isoclinism | isoclinic class of C3⋊AGL(1,𝔽7) |
| automorphism | Aut(G) = D32×AGL(1,𝔽7) (order 1512) · Inn(G) = C3⋊AGL(1,𝔽7) · Out(G) = C2×D3 |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C3 |
| Fitting, socle | C32×C7 ⋊ C2×C3 |
| derived | C3×C7 ⋊ C2×C32 |
| Hall | C32×ΓL(1,𝔽8) ⋊ C2 |
| Sylow | C7 ⋊ C32×D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C3⋊AGL(1,𝔽7) |
| commutator, cosocle, hypocenter | C3×C7 | C2×C32 |
| Frattini | C1 | G |
| Fitting, socle | C32×C7 | C2×C3 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C3 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 42 |
A minimal faithful permutation representation — 2 generators on 13 points:
(5,6)(8,11,12,9,10,13);(1,2,3)(4,5,6)(7,8,10,13,12,11,9)36 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(378, 49);