Finite Group DatabasebetaFinite Group Databasebeta| order | 138 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×D3×C11 (order 132) · Inn(G) = D3 · Out(G) = C2×C11 |
| holomorph | G ⋊ Aut(G) = D32×AGL(1,𝔽23) |
| homology | Abelianization (H1) = C2×C23 · Schur multiplier (H2) = C1 |
| direct product | D3×C23 |
| unambiguous semidirect product | C3⋊(C2×C23) |
| Fitting, socle, Hall | C3×C23 ⋊ C2 |
| derived, Sylow | C3 ⋊ C2×C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23 | D3 |
| commutator, cosocle, hypocenter | C3 | C2×C23 |
| Frattini | C1 | G |
| Fitting, socle | C3×C23 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C23 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 69 |
A minimal faithful permutation representation — 2 generators on 26 points:
(2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26);(1,2,3)69 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D23, C2×D3×C23, D9×C23, … |
| maximal normal subgroup of | D3×D23, C2×D3×C23, C3×D3×C23, … |
| maximal quotient of | Dic3×C23, C2×D3×C23, D9×C23, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(138, 1);