Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of #66561 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C3×C7 ⋊ Q16 |
| Sylow | C7 ⋊ #216 |
| Sylow | C3 ⋊ Dic28 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4246 |
| commutator, Fitting, cosocle | C3×C4×C7 | C22 |
| Frattini, hypercenter | C4 | D3×D7 |
| socle, prime radical | C2×C3×C7 | D4 |
| hypocenter | C3×C7 | Q16 |
| LCS | G ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ C4 | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 2 generators on 26 points:
(1,2,4,6,7,5,3)(9,10)(11,13,20,24,15,22,14,21)(12,16,23,26,18,25,17,19);(2,3)(4,5)(6,7)(8,9,10)(11,12,15,18)(13,19,22,26)(14,23,20,17)(16,24,25,21)Polycyclic presentation:
2^4,3,7|a^2=d,b^2=cd,c^2=d,b^a=bc,c^a=cd,f^a=f^6,e^b=e^248 conjugacy classes — view the full character table.
| maximal subgroup of | D3×Dic28, #10620825, #10620834, … |
| maximal normal subgroup of | D3×Dic28, #10620825, #10620834, … |
| maximal quotient of | #10620506, #10620509, #10620525, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 39);