Finite Group DatabasebetaFinite Group Databasebeta| order | 312 |
| isoclinism | isoclinic class of C22⋊(C13⋊C3) |
| automorphism | Aut(G) = A4×AGL(1,𝔽13) (order 1872) · Inn(G) = C22⋊(C13⋊C3) · Out(G) = C3×C4 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×C22⋊(C13⋊C3) |
| unambiguous semidirect product | C23⋊(C13⋊C3) |
| unambiguous semidirect product | (C2×C13)⋊A4 |
| Fitting, socle, Hall | C23×C13 ⋊ C3 |
| derived | C22×C13 ⋊ C2×C3 |
| Sylow | C13 ⋊ C2×A4 |
| Sylow | C23 ⋊ C13⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C22⋊(C13⋊C3) |
| commutator, cosocle, hypocenter | C22×C13 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C23×C13 | C3 |
| LCS | G ▷ C22×C13 | |
| UCS | C1 ◁ C2 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 78 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2)(4,5,6)(8,9,11)(10,13,16)(12,15,14)(17,19,18);(3,6)(4,5)(7,8,10,9,12,16,13,17,19,11,14,18,15)40 conjugacy classes — view the full character table.
| maximal subgroup of | Dic13⋊A4, C4×C22⋊(C13⋊C3), C2×D13⋊A4, … |
| maximal normal subgroup of | Dic13⋊A4, C4×C22⋊(C13⋊C3), C2×D13⋊A4, … |
| maximal quotient of | C4×C22⋊(C13⋊C3), C2×Q8⋊(C13⋊C3), #41⋊(C13⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(312, 57);