Finite Group DatabasebetaFinite Group Databasebeta| order | 312 |
| isoclinism | isoclinic class of C13⋊C3 |
| automorphism | Aut(G) = C22×AGL(1,𝔽13) (order 624) · Inn(G) = C13⋊C3 · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C3×C8 · Schur multiplier (H2) = C1 |
| direct product | C8×C13⋊C3 |
| unambiguous semidirect product | (C8×C13)⋊C3 |
| Fitting, Hall | C8×C13 ⋊ C3 |
| derived, Sylow | C13 ⋊ C3×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C8 | C13⋊C3 |
| commutator, hypocenter | C13 | C3×C8 |
| Frattini | C4 | C2×C13⋊C3 |
| Fitting | C8×C13 | C3 |
| socle | C2×C13 | C3×C4 |
| prime radical | C2×C13⋊C3 | C4 |
| cosocle | C4×C13 | C2×C3 |
| LCS | G ▷ C13 | |
| UCS | C1 ◁ C8 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 104 |
A minimal faithful permutation representation — 2 generators on 21 points:
(2,3,5)(4,7,10)(6,9,8)(11,13,12)(14,15,16,18,17,19,20,21);(1,2,4,3,6,10,7,11,13,5,8,12,9)56 conjugacy classes — view the full character table.
| maximal subgroup of | #6090⋊C3, C16×C13⋊C3, C8×D13⋊C3, … |
| maximal normal subgroup of | #6090⋊C3, C16×C13⋊C3, C8×D13⋊C3, … |
| maximal quotient of | C16×C13⋊C3, C2×C8×C13⋊C3, C8×C13⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(312, 2);