Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of Dic14 |
| automorphism | Aut(G) = C22×D4×AGL(1,𝔽7) (order 1344) · Inn(G) = C2×D7 · Out(G) = C24×C3 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#1157 |
| Fitting | C2×C3×C4×C7 ⋊ C2 |
| Hall | C3×C7 ⋊ #41 |
| Sylow | C7 ⋊ C3×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×D7 |
| commutator, cosocle | C2×C7 | C23×C3 |
| Frattini | C2 | C22×C3×D7 |
| Fitting | C2×C3×C4×C7 | C2 |
| socle | C2×C3×C7 | C23 |
| hypercenter | C2×C3×C4 | D7 |
| hypocenter | C7 | C3×#41 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×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)(9,13)(12,15)(16,17,18);(8,14,11,10)(9,15,13,12);(2,3)(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)102 conjugacy classes — view the full character table.
| maximal subgroup of | #10620511, #10620512, C3×#6379, … |
| maximal normal subgroup of | #10620511, #10620512, C3×#6379, … |
| maximal quotient of | C3×C4×Dic14, C3×#6433, C3×#6435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 177);