Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of #66561 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C3×C7 ⋊ D8 |
| Sylow | C7 ⋊ #213 |
| Sylow | C3 ⋊ D56 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4246 |
| commutator, Fitting, cosocle | C3×C4×C7 | C22 |
| Frattini, hypercenter | C4 | D3×D7 |
| socle | C2×C3×C7 | D4 |
| hypocenter | C3×C7 | D8 |
| LCS | G ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ C4 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,4,6,7,5,3)(9,10)(11,12,16,18,14,17,13,15);(2,3)(4,5)(6,7)(8,9,10)(12,15)(13,16)(17,18)Polycyclic presentation:
2^4,3,7|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×D56, #10620815, #10620816, … |
| maximal normal subgroup of | D3×D56, #10620815, #10620816, … |
| maximal quotient of | #10620473, #10620477, #10620478, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 30);