Finite Group DatabasebetaFinite Group Databasebeta| order | 312 |
| isoclinism | isoclinic class of C13⋊C3 |
| automorphism | Aut(G) = D4×AGL(1,𝔽13) (order 1248) · Inn(G) = C13⋊C3 · Out(G) = C4×D4 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×C4×C13⋊C3 |
| unambiguous semidirect product | (C2×C4×C13)⋊C3 |
| Fitting, Hall | C2×C4×C13 ⋊ C3 |
| derived, Sylow | C13 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C13⋊C3 |
| commutator, hypocenter | C13 | C2×C3×C4 |
| Frattini | C2 | C22×C13⋊C3 |
| Fitting | C2×C4×C13 | C3 |
| socle | C22×C13 | C2×C3 |
| prime radical | C22×C13⋊C3 | C2 |
| cosocle | C2×C13 | C22×C3 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C2×C4 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 104 |
A minimal faithful permutation representation — 2 generators on 19 points:
(3,4,5,6)(8,9,11)(10,13,16)(12,15,14)(17,19,18);(1,2)(7,8,10,9,12,16,13,17,19,11,14,18,15)56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1055⋊C3, #6099⋊C3, C4×Dic13⋊C3, … |
| maximal normal subgroup of | C2×#1055⋊C3, #6099⋊C3, C4×Dic13⋊C3, … |
| maximal quotient of | C42×C13⋊C3, C22⋊C4×C13⋊C3, C4⋊C4×C13⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(312, 22);