Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)⋊Q8 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽7) (order 672) · Inn(G) = C2×AGL(1,𝔽7) · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | #1136⋊C3 |
| Hall | #1136 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ M16 |
| Sylow | C7 ⋊ C3×M16 |
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)(9,12)(13,15);(2,3)(4,6)(5,7)(8,9,10,13,11,12,14,15)Polycyclic presentation:
2^2,3,2^2,7|a^2=d,d^2=e,b^a=be,f^a=f^6,f^c=f^438 conjugacy classes — view the full character table.
| maximal subgroup of | #6379⋊C3, #6393⋊C3, #6396⋊C3, … |
| maximal normal subgroup of | #6379⋊C3, #6393⋊C3, #6396⋊C3, … |
| maximal quotient of | #6377⋊C3, C4⋊(ΓL(1,𝔽8)⋊C8), (C22⋊(C7⋊C8))⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 13);