Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of #4245 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C3×C7 ⋊ M16 |
| Sylow | C7 ⋊ ΓL(1,𝔽25) |
| Sylow | C3 ⋊ #1136 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D3×D7 |
| commutator, socle | C2×C3×C7 | C2×C4 |
| Fitting, cosocle | C3×C4×C7 | C22 |
| prime radical | C2×D3×C7 | C4 |
| hypocenter | C3×C7 | M16 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C4 | |
40 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)(12,15)(16,18);(2,3)(4,5)(6,7)(8,9,10)(11,12,13,16,14,15,17,18)Polycyclic presentation:
2^4,3,7|a^2=c,c^2=d,b^a=bd,f^a=f^6,e^b=e^254 conjugacy classes — view the full character table.
| maximal subgroup of | D7×ΓL(1,𝔽25), #10620808, #10620827, … |
| maximal normal subgroup of | D7×ΓL(1,𝔽25), #10620808, #10620827, … |
| maximal quotient of | #10620488, (C2×D3)⋊(C7⋊C8), Dic3⋊(C7⋊C8), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 27);