Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×C7×Q8 ⋊ C2 |
| Hall | C3×C7 ⋊ #41 |
| Sylow | C7 ⋊ #239 |
| Sylow | C3 ⋊ #1161 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D21 |
| commutator, socle, cosocle | C2×C3×C7 | C23 |
| Fitting | C3×C7×Q8 | C2 |
| hypercenter | Q8 | D21 |
| hypocenter | C3×C7 | #41 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2,4,6,7,5,3)(8,9,10)(11,13,14,17)(12,15,16,18);(11,12,14,16)(13,18,17,15);(2,3)(4,5)(6,7)(9,10)(13,17)(15,18)Polycyclic presentation:
2^4,3,7|b^2=d,c^2=d,c^a=cd,e^a=e^2,f^a=f^6,c^b=cd60 conjugacy classes — view the full character table.
| maximal subgroup of | #10621068, #10621074, #10621083, … |
| maximal normal subgroup of | #10621068, #10621074, #10621083, … |
| maximal quotient of | #10621373, #10621375, #10621376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 201);