Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of #1141⋊C3 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽7) (order 672) · Inn(G) = (C22⋊D7)⋊C3 · Out(G) = C22 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #1144⋊C3 |
| Hall | #1144 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ Q16 |
| Sylow | C7 ⋊ C3×Q16 |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,7)(3,4,6)(8,10,12,18)(9,13,15,22)(11,20,21,17)(14,23,19,16);(2,3)(4,6)(5,7)(8,9,12,15)(10,16,18,23)(11,19,21,14)(13,20,22,17)Polycyclic presentation:
2^2,3,2^2,7|a^2=e,b^2=e,d^2=e,b^a=bd,d^a=de,f^a=f^6,d^b=de,f^c=f^426 conjugacy classes — view the full character table.
| maximal subgroup of | #6478⋊C3, #6479⋊C3, Q16×AGL(1,𝔽7), … |
| maximal normal subgroup of | #6478⋊C3, #6479⋊C3, Q16×AGL(1,𝔽7), … |
| maximal quotient of | #6381⋊C3, #6384⋊C3, #6409⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 21);