Finite Group DatabasebetaFinite Group Databasebeta| order | 378 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | Aut(G) = C9×AGL(1,𝔽7) (order 378) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C2×C9 |
| homology | Abelianization (H1) = C2×C27 · Schur multiplier (H2) = C1 |
| direct product | C2×C7⋊C27 |
| unambiguous semidirect product | (C2×C7)⋊C27 |
| derived, Sylow | C7 ⋊ C2×C27 |
| Hall | C2×C7 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C9 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C2×C27 |
| Frattini | C9 | C2×ΓL(1,𝔽8) |
| Fitting | C2×C7×C9 | C3 |
| socle, prime radical | C2×C3×C7 | C9 |
| cosocle | C7×C9 | C2×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C9 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 378 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2)(4,5,6)(7,9,8)(10,15,27,18,30,29,32,11,20,13,23,33,26,35,14,17,16,28,19,31,36,12,22,21,25,24,34);(3,4,5,7,6,8,9)90 conjugacy classes — view the full character table.
| maximal subgroup of | Dic7⋊C27, C4×C7⋊C27, C2×D7⋊C27, … |
| maximal normal subgroup of | Dic7⋊C27, C4×C7⋊C27, C2×D7⋊C27, … |
| maximal quotient of | C4×C7⋊C27, C22×C7⋊C27, C2×C7⋊C81, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(378, 2);