Finite Group DatabasebetaFinite Group Databasebeta| order | 312 |
| isoclinism | isoclinic class of C3⋊#261 |
| automorphism | Aut(G) = C2×D3×AGL(1,𝔽13) (order 1872) · Inn(G) = C3⋊#261 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | C2×C3⋊#261 |
| unambiguous semidirect product | (C2×C3)⋊#261 |
| unambiguous semidirect product | D13⋊Dic3 |
| Fitting, socle | C2×C3×C13 ⋊ C4 |
| derived, Hall | C3×C13 ⋊ C2×C4 |
| Sylow | C13 ⋊ C2×Dic3 |
| Sylow | C3 ⋊ C2×#261 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | C3⋊#261 |
| commutator, hypocenter | C3×C13 | C2×C4 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C13 | C4 |
| prime radical | C2×C3×D13 | C2 |
| cosocle | C3×D13 | C22 |
| LCS | G ▷ C3×C13 | |
| UCS | C1 ◁ C2 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 78 |
A minimal faithful permutation representation — 2 generators on 18 points:
(4,5)(7,8,9,11)(10,13,14,15)(12,17,18,16);(1,2)(3,4,5)(6,7,10,15,17,8,12,18,11,16,13,14,9)30 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×#261, (C2×D3)⋊#261, Dic3⋊#261, … |
| maximal normal subgroup of | Dic3×#261, (C2×D3)⋊#261, Dic3⋊#261, … |
| maximal quotient of | D13⋊(C3⋊C8), #10597724, C4×C3⋊#261, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(312, 53);