Finite Group DatabasebetaFinite Group Databasebeta| order | 78 |
| isoclinism | isoclinic class of C13⋊C3 |
| automorphism | Aut(G) = AGL(1,𝔽13) (order 156) · Inn(G) = C13⋊C3 · Out(G) = C4 |
| holomorph | G ⋊ Aut(G) = C2×#14523144⋊C3 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| general semilinear | ΓL(1,𝔽27) |
| direct product | C2×C13⋊C3 |
| unambiguous semidirect product | (C2×C13)⋊C3 |
| Fitting, socle, Hall | C2×C13 ⋊ C3 |
| derived, Sylow | C13 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C13⋊C3 |
| commutator, cosocle, hypocenter | C13 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C13 | C3 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C2 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 26 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2)(4,5,7)(6,9,12)(8,11,10)(13,15,14);(3,4,6,5,8,12,9,13,15,7,10,14,11)14 conjugacy classes — view the full character table.
| maximal subgroup of | Dic13⋊C3, C4×C13⋊C3, C2×D13⋊C3, … |
| maximal normal subgroup of | Dic13⋊C3, C4×C13⋊C3, C2×D13⋊C3, … |
| maximal quotient of | C4×C13⋊C3, C22×C13⋊C3, C2×C13⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(78, 2);