Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of #1132⋊C3 |
| automorphism | Aut(G) = AGL(1,ℤ/8)×AGL(1,𝔽7) (order 1344) · Inn(G) = C4⋊AGL(1,𝔽7) · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | Dic28⋊C3 |
| Hall | Dic28 ⋊ 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)(8,10,17,21,12,19,11,18)(9,13,20,23,15,22,14,16);(2,3)(4,6)(5,7)(8,9,12,15)(10,16,19,23)(11,20,17,14)(13,21,22,18)Polycyclic presentation:
2^2,3,2^2,7|a^2=e,b^2=de,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 | #6374⋊C3, Dic56⋊C3, #6401⋊C3, … |
| maximal normal subgroup of | #6374⋊C3, Dic56⋊C3, #6401⋊C3, … |
| maximal quotient of | #6390⋊C3, #6392⋊C3, C2×Dic28⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 11);