Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of #1132⋊C3 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽7) (order 672) · Inn(G) = C4⋊AGL(1,𝔽7) · Out(G) = C22 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #1132⋊C3 |
| Fitting | C7×C8 ⋊ C2×C3 |
| Hall | #1132 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ SD16 |
| Sylow | C7 ⋊ C3×SD16 |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,7)(3,4,6)(8,9,10,12,11,13,14,15);(2,3)(4,6)(5,7)(9,12)(10,14)(13,15)Polycyclic presentation:
2^2,3,2^2,7|b^2=d,d^2=e,b^a=bd,d^a=de,f^a=f^6,f^c=f^429 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#1132⋊C3, #6467⋊C3, #6471⋊C3, … |
| maximal normal subgroup of | C2×#1132⋊C3, #6467⋊C3, #6471⋊C3, … |
| maximal quotient of | #6390⋊C3, #6391⋊C3, #6395⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 9);