Finite Group DatabasebetaFinite Group Databasebeta| order | 336 |
| isoclinism | isoclinic class of Dic42 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | C3×C7 ⋊ C22⋊C4 |
| Sylow | C7 ⋊ #217 |
| Sylow | C3 ⋊ C22⋊Dic7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D21 |
| commutator | C2×C3×C7 | C2×C4 |
| Fitting, prime radical | C23×C3×C7 | C2 |
| socle, cosocle | C22×C3×C7 | C22 |
| hypercenter | C23 | D21 |
| hypocenter | C3×C7 | C22⋊C4 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C23 | |
68 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)(16,18);(2,3)(4,5)(6,7)(9,10)(11,12,13,14)(15,16)(17,18)Polycyclic presentation:
2^4,3,7|a^2=c,b^a=bd,e^a=e^2,f^a=f^690 conjugacy classes — view the full character table.
| maximal subgroup of | #10620530, #10620531, #10620645, … |
| maximal normal subgroup of | #10620530, #10620531, #10620645, … |
| maximal quotient of | #10620669, #10620670, #10620671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(336, 105);