Finite Group DatabasebetaFinite Group Databasebeta| order | 195 |
| isoclinism | isoclinic class of C13⋊C3 |
| automorphism | Aut(G) = C4×AGL(1,𝔽13) (order 624) · Inn(G) = C13⋊C3 · Out(G) = C42 |
| homology | Abelianization (H1) = C3×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×C13⋊C3 |
| unambiguous semidirect product | C13⋊(C3×C5) |
| unambiguous semidirect product | (C5×C13)⋊C3 |
| Fitting, socle, Hall | C5×C13 ⋊ C3 |
| derived, Sylow | C13 ⋊ C3×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | C13⋊C3 |
| commutator, cosocle, hypocenter | C13 | C3×C5 |
| Frattini | C1 | G |
| Fitting, socle | C5×C13 | C3 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C5 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 65 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4,5)(7,8,10)(9,12,15)(11,14,13)(16,18,17);(6,7,9,8,11,15,12,16,18,10,13,17,14)35 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D13⋊C3, D5×C13⋊C3, C5⋊(D13⋊C3), … |
| maximal normal subgroup of | C5×D13⋊C3, D5×C13⋊C3, C5⋊(D13⋊C3), … |
| maximal quotient of | C2×C5×C13⋊C3, C5×C13⋊C9, C3×C5×C13⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(195, 1);