Finite Group DatabasebetaFinite Group Databasebeta| order | 312 |
| isoclinism | isoclinic class of D13⋊C3 |
| automorphism | Aut(G) = C22×AGL(1,𝔽13) (order 624) · Inn(G) = D13⋊C3 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C4×D13⋊C3 |
| unambiguous semidirect product | (C4×D13)⋊C3 |
| Fitting | C4×C13 ⋊ C2×C3 |
| derived, Sylow | C13 ⋊ C2×C3×C4 |
| Hall | C4×D13 ⋊ C3 |
| Hall | C13⋊C3 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D13⋊C3 |
| commutator, hypocenter | C13 | C2×C3×C4 |
| Frattini | C2 | C2×D13⋊C3 |
| Fitting | C4×C13 | C2×C3 |
| socle, cosocle | C2×C13 | C22×C3 |
| prime radical | C2×D13⋊C3 | C2 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C4 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 52 |
A minimal faithful permutation representation — 2 generators on 17 points:
(2,6,7,3,4,8)(5,12,11,9,10,13);(1,2,5,4,8,11,10,12,13,7,6,9,3)(14,15,16,17)32 conjugacy classes — view the full character table.
| maximal subgroup of | C8×D13⋊C3, #6094⋊C3, (D13⋊C3)⋊C8, … |
| maximal normal subgroup of | C8×D13⋊C3, #6094⋊C3, (D13⋊C3)⋊C8, … |
| maximal quotient of | C8×D13⋊C3, #6094⋊C3, C4×Dic13⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(312, 9);