Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | C3×C7 ⋊ #41 |
| Sylow | C7 ⋊ #237 |
| Sylow | C3 ⋊ #1157 |
| 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 | #41 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ C4 | |
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)(11,14,15,13)(12,16,17,18);(9,10)(11,13,15,14)(12,16,17,18);(2,3)(4,5)(6,7)(11,12)(13,16)(14,18)(15,17)Polycyclic presentation:
2^4,3,7|b^2=d,c^2=d,c^a=cd,f^a=f^6,e^b=e^251 conjugacy classes — view the full character table.
| maximal subgroup of | #10620816, #10620834, #10620835, … |
| maximal normal subgroup of | #10620816, #10620834, #10620835, … |
| maximal quotient of | #10620900, #10620901, #10620902, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 138);