Finite Group DatabasebetaFinite Group Databasebeta| order | 378 |
| isoclinism | isoclinic class of #67355 |
| automorphism | Aut(G) = H(3,𝔽3)×AGL(1,𝔽7) (order 1134) · Inn(G) = C3×AGL(1,𝔽7) · Out(G) = C32 |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #4430⋊C2 |
| Hall | #4430 ⋊ 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 | C32×C7 | C2×C3 |
| prime radical | C32×D7 | C3 |
| hypercenter | C32 | AGL(1,𝔽7) |
| hypocenter | C7 | C2×3−1+2 |
| LCS | G ▷ C3×C7 ▷ C7 | |
| UCS | C1 ◁ C3 ◁ C32 | |
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)(9,12,13)(11,15,16);(2,4,5)(3,6,7)(8,13,16,10,12,11,14,9,15)Polycyclic presentation:
2,3^3,7|b^3=d,e^a=e^6,c^b=cd,e^b=e^431 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#4430⋊C2, #11765586, #10588039⋊C2, … |
| maximal normal subgroup of | C2×#4430⋊C2, #11765586, #10588039⋊C2, … |
| maximal quotient of | #4430⋊C4, C2×#4430⋊C2, (D7⋊C9)⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(378, 11);