Finite Group DatabasebetaFinite Group Databasebeta| order | 156 |
| isoclinism | isoclinic class of C13⋊C3 |
| automorphism | Aut(G) = C2×AGL(1,𝔽13) (order 312) · Inn(G) = C13⋊C3 · Out(G) = C2×C4 |
| holomorph | G ⋊ Aut(G) = D4×#14523144⋊C3 |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C1 |
| direct product | C4×C13⋊C3 |
| unambiguous semidirect product | (C4×C13)⋊C3 |
| Fitting, Hall | C4×C13 ⋊ C3 |
| derived, Sylow | C13 ⋊ C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C13⋊C3 |
| commutator, hypocenter | C13 | C3×C4 |
| Frattini | C2 | C2×C13⋊C3 |
| Fitting | C4×C13 | C3 |
| socle, cosocle | C2×C13 | C2×C3 |
| prime radical | C2×C13⋊C3 | C2 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C4 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 52 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,3,5)(4,7,10)(6,9,8)(11,13,12)(14,15,16,17);(1,2,4,3,6,10,7,11,13,5,8,12,9)28 conjugacy classes — view the full character table.
| maximal subgroup of | #1055⋊C3, C8×C13⋊C3, (C13⋊C3)⋊Q8, … |
| maximal normal subgroup of | #1055⋊C3, C8×C13⋊C3, (C13⋊C3)⋊Q8, … |
| maximal quotient of | C8×C13⋊C3, C2×C4×C13⋊C3, C4×C13⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(156, 2);