Finite Group DatabasebetaFinite Group Databasebeta| order | 378 |
| isoclinism | isoclinic class of #67355 |
| automorphism | Aut(G) = C3×D3×AGL(1,𝔽7) (order 756) · Inn(G) = C3×AGL(1,𝔽7) · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C7×C9 ⋊ C2×C3 |
| Hall | #4427 ⋊ C2 |
| Hall | D7 ⋊ 3−1+2 |
| Sylow | C7 ⋊ C2×3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×AGL(1,𝔽7) |
| commutator, socle, cosocle | C3×C7 | C2×C32 |
| Fitting | C7×C9 | C2×C3 |
| prime radical | C3×AGL(1,𝔽7) | C3 |
| hypercenter | C9 | AGL(1,𝔽7) |
| hypocenter | C7 | C2×3−1+2 |
| LCS | G ▷ C3×C7 ▷ C7 | |
| UCS | C1 ◁ C3 ◁ C9 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 63 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2)(3,5)(6,7)(8,14,15,13,11,12,10,9,16);(2,4,5)(3,6,7)(9,11,14)(12,16,15)Polycyclic presentation:
2,3^3,7|c^3=d^2,e^a=e^6,c^b=cd,e^b=e^431 conjugacy classes — view the full character table.
| maximal subgroup of | #67356⋊C2, C2×#67356, C3×#67356, … |
| maximal normal subgroup of | #67356⋊C2, C2×#67356, C3×#67356, … |
| maximal quotient of | #10625387, C2×#67356, (D7⋊C9)⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(378, 9);