Finite Group DatabasebetaFinite Group Databasebeta| order | 378 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C9 · Schur multiplier (H2) = C3 |
| direct product | C2×C3×C7⋊C9 |
| unambiguous semidirect product | (C2×C3×C7)⋊C9 |
| derived, Sylow | C7 ⋊ C2×C3×C9 |
| Hall | C2×C7 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C32 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C2×C3×C9 |
| Frattini | C3 | C2×C3×ΓL(1,𝔽8) |
| Fitting, socle, prime radical | C2×C32×C7 | C3 |
| cosocle | C3×C7 | C2×C32 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C32 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 378 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2)(7,8,9)(10,12,11)(13,17,21,15,20,16,18,14,19);(3,4,5)(6,7,8,10,9,11,12)90 conjugacy classes — view the full character table.
| maximal subgroup of | C3×Dic7⋊C9, Dic3×C7⋊C9, C3⋊(Dic7⋊C9), … |
| maximal normal subgroup of | C3×Dic7⋊C9, Dic3×C7⋊C9, C3⋊(Dic7⋊C9), … |
| maximal quotient of | C3×C4×C7⋊C9, C22×C3×C7⋊C9, C2×C3×C7⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(378, 26);