Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of #66632 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C3×C7×D4 ⋊ C2 |
| Hall | C3×C7 ⋊ D8 |
| Sylow | C7 ⋊ #213 |
| Sylow | C3 ⋊ #1141 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #4268 |
| commutator, cosocle | C3×C4×C7 | C22 |
| Frattini | C4 | C2×D21 |
| Fitting | C3×C7×D4 | C2 |
| socle | C2×C3×C7 | D4 |
| hypercenter | D4 | D21 |
| hypocenter | C3×C7 | D8 |
| LCS | G ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
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)(8,9,10)(12,15)(13,18)(16,17);(2,3)(4,5)(6,7)(9,10)(11,12)(13,15)(14,17)(16,18)Polycyclic presentation:
2^4,3,7|c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^6,c^b=cd57 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#213, #10621040, D3×#1141, … |
| maximal normal subgroup of | D7×#213, #10621040, D3×#1141, … |
| maximal quotient of | #10620646, #10620648, #10620665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 101);