Finite Group DatabasebetaFinite Group Databasebeta| order | 117 |
| isoclinism | isoclinic class of C13⋊C3 |
| automorphism | Aut(G) = D3×AGL(1,𝔽13) (order 936) · Inn(G) = C13⋊C3 · Out(G) = C4×D3 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C3 |
| direct product | C3×C13⋊C3 |
| unambiguous semidirect product | C13⋊C32 |
| unambiguous semidirect product | (C3×C13)⋊C3 |
| Fitting, socle | C3×C13 ⋊ C3 |
| derived, Sylow | C13 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | C13⋊C3 |
| commutator, cosocle, hypocenter | C13 | C32 |
| Frattini | C1 | G |
| Fitting, socle | C3×C13 | C3 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C3 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 39 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,3,5)(4,7,10)(6,9,8)(11,13,12);(1,2,4,3,6,10,7,11,13,5,8,12,9)(14,15,16)21 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D13⋊C3, D3×C13⋊C3, C3⋊(D13⋊C3), … |
| maximal normal subgroup of | C3×D13⋊C3, D3×C13⋊C3, C3⋊(D13⋊C3), … |
| maximal quotient of | C2×C3×C13⋊C3, C9×C13⋊C3, #66848, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(117, 3);