Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×C4×C7 ⋊ C22 |
| Hall | C3×C7 ⋊ C2×D4 |
| Sylow | C7 ⋊ D3×D4 |
| Sylow | C3 ⋊ D4×D7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×D7 |
| commutator, socle, cosocle | C2×C3×C7 | C23 |
| Fitting | C3×C4×C7 | C22 |
| hypercenter | C4 | D3×D7 |
| hypocenter | C3×C7 | C2×D4 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ C4 | |
108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 3 generators on 14 points:
(1,2,4,6,7,5,3)(8,9,10)(11,12,13,14);(9,10)(12,14);(2,3)(4,5)(6,7)(12,14)Polycyclic presentation:
2^4,3,7|c^2=d,c^a=cd,f^a=f^6,c^b=cd,e^b=e^242 conjugacy classes — view the full character table.
| maximal subgroup of | #10620817, #10620818, #10620819, … |
| maximal normal subgroup of | #10620817, #10620818, #10620819, … |
| maximal quotient of | #10620817, #10620818, #10620819, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 150);