Finite Group DatabasebetaFinite Group Databasebeta| order | 156 |
| isoclinism | isoclinic class of D13⋊C3 |
| automorphism | Aut(G) = C2×AGL(1,𝔽13) (order 312) · Inn(G) = D13⋊C3 · Out(G) = C22 |
| holomorph | G ⋊ Aut(G) = #72242057 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×D13⋊C3 |
| unambiguous semidirect product | D13⋊(C2×C3) |
| unambiguous semidirect product | (C13⋊C3)⋊C22 |
| unambiguous semidirect product | (C2×D13)⋊C3 |
| unambiguous semidirect product | (C2×C13⋊C3)⋊C2 |
| Fitting, socle | C2×C13 ⋊ C2×C3 |
| derived, Sylow | C13 ⋊ C22×C3 |
| Hall | C2×D13 ⋊ C3 |
| Hall | C13⋊C3 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D13⋊C3 |
| commutator, cosocle, hypocenter | C13 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C13 | C2×C3 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C2 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 26 |
A minimal faithful permutation representation — 2 generators on 15 points:
(4,8,9,5,6,10)(7,14,13,11,12,15);(1,2)(3,4,7,6,10,13,12,14,15,9,8,11,5)16 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D13⋊C3, C4⋊(D13⋊C3), (C22⋊D13)⋊C3, … |
| maximal normal subgroup of | C4×D13⋊C3, C4⋊(D13⋊C3), (C22⋊D13)⋊C3, … |
| maximal quotient of | (C13⋊C3)⋊Q8, C4×D13⋊C3, C4⋊(D13⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(156, 8);