Finite Group DatabasebetaFinite Group Databasebeta| order | 312 |
| isoclinism | isoclinic class of D3×D13 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×D3×D13 |
| unambiguous semidirect product | (D3×C13)⋊C22 |
| unambiguous semidirect product | (C3×D13)⋊C22 |
| unambiguous semidirect product | D39⋊C22 |
| Fitting, socle | C2×C3×C13 ⋊ C22 |
| derived, Hall | C3×C13 ⋊ C23 |
| Sylow | C13 ⋊ C22×D3 |
| Sylow | C3 ⋊ C22×D13 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×D13 |
| commutator, cosocle, hypocenter | C3×C13 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C13 | C22 |
| LCS | G ▷ C3×C13 | |
| UCS | C1 ◁ C2 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 78 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2)(3,4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18);(4,5)(6,7,9,11,13,15,17,18,16,14,12,10,8);(1,2)48 conjugacy classes — view the full character table.
| maximal subgroup of | (C2×D3)⋊#261, C4×D3×D13, D12×D13, … |
| maximal normal subgroup of | (C2×D3)⋊#261, C4×D3×D13, D12×D13, … |
| maximal quotient of | Dic6×D13, #10597731, #10597732, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(312, 54);