Finite Group DatabasebetaFinite Group Databasebeta| order | 378 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)⋊D9 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊D9 |
| Fitting, derived | C7×C9 ⋊ C2×C3 |
| Hall | C9×ΓL(1,𝔽8) ⋊ C2 |
| Sylow | C7 ⋊ C3×D9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C7×C9 | C2×C3 |
| Frattini | C3 | C3⋊AGL(1,𝔽7) |
| socle | C3×C7 | C3×D3 |
| LCS | G ▷ C7×C9 | |
| UCS | C1 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 63 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,5,6,3,4,7)(9,11)(10,14)(12,13)(15,16);(1,2,4,7,6,5,3)(8,16,11,14,13,12,10,9,15)Polycyclic presentation:
2,3^2,7,3|c^3=e^2,c^a=c^2e,d^a=d^6,e^a=e^2,d^b=d^227 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(378, 20);