Copied to
clipboard

G = S3×C7order 42 = 2·3·7

Direct product of C7 and S3

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: S3×C7, C3⋊C14, C213C2, SmallGroup(42,3)

Series: Derived Chief Lower central Upper central

C1C3 — S3×C7
C1C3C21 — S3×C7
C3 — S3×C7
C1C7

Generators and relations for S3×C7
 G = < a,b,c | a7=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >

3C2
3C14

Character table of S3×C7

 class 1237A7B7C7D7E7F14A14B14C14D14E14F21A21B21C21D21E21F
 size 132111111333333222222
ρ1111111111111111111111    trivial
ρ21-11111111-1-1-1-1-1-1111111    linear of order 2
ρ31-11ζ75ζ73ζ74ζ7ζ76ζ7276772737475ζ7ζ75ζ73ζ76ζ72ζ74    linear of order 14
ρ4111ζ74ζ7ζ76ζ75ζ72ζ73ζ72ζ75ζ73ζ7ζ76ζ74ζ75ζ74ζ7ζ72ζ73ζ76    linear of order 7
ρ51-11ζ72ζ74ζ73ζ76ζ7ζ7577675747372ζ76ζ72ζ74ζ7ζ75ζ73    linear of order 14
ρ6111ζ76ζ75ζ72ζ74ζ73ζ7ζ73ζ74ζ7ζ75ζ72ζ76ζ74ζ76ζ75ζ73ζ7ζ72    linear of order 7
ρ71-11ζ74ζ7ζ76ζ75ζ72ζ7372757377674ζ75ζ74ζ7ζ72ζ73ζ76    linear of order 14
ρ81-11ζ76ζ75ζ72ζ74ζ73ζ773747757276ζ74ζ76ζ75ζ73ζ7ζ72    linear of order 14
ρ91-11ζ7ζ72ζ75ζ73ζ74ζ7674737672757ζ73ζ7ζ72ζ74ζ76ζ75    linear of order 14
ρ101-11ζ73ζ76ζ7ζ72ζ75ζ7475727476773ζ72ζ73ζ76ζ75ζ74ζ7    linear of order 14
ρ11111ζ73ζ76ζ7ζ72ζ75ζ74ζ75ζ72ζ74ζ76ζ7ζ73ζ72ζ73ζ76ζ75ζ74ζ7    linear of order 7
ρ12111ζ72ζ74ζ73ζ76ζ7ζ75ζ7ζ76ζ75ζ74ζ73ζ72ζ76ζ72ζ74ζ7ζ75ζ73    linear of order 7
ρ13111ζ75ζ73ζ74ζ7ζ76ζ72ζ76ζ7ζ72ζ73ζ74ζ75ζ7ζ75ζ73ζ76ζ72ζ74    linear of order 7
ρ14111ζ7ζ72ζ75ζ73ζ74ζ76ζ74ζ73ζ76ζ72ζ75ζ7ζ73ζ7ζ72ζ74ζ76ζ75    linear of order 7
ρ1520-1222222000000-1-1-1-1-1-1    orthogonal lifted from S3
ρ1620-17477675727300000075747727376    complex faithful
ρ1720-17675727473700000074767573772    complex faithful
ρ1820-17376772757400000072737675747    complex faithful
ρ1920-17727573747600000073772747675    complex faithful
ρ2020-17274737677500000076727477573    complex faithful
ρ2120-17573747767200000077573767274    complex faithful

Permutation representations of S3×C7
On 21 points - transitive group 21T6
Generators in S21
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)
(1 10 18)(2 11 19)(3 12 20)(4 13 21)(5 14 15)(6 8 16)(7 9 17)
(8 16)(9 17)(10 18)(11 19)(12 20)(13 21)(14 15)

G:=sub<Sym(21)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,10,18)(2,11,19)(3,12,20)(4,13,21)(5,14,15)(6,8,16)(7,9,17), (8,16)(9,17)(10,18)(11,19)(12,20)(13,21)(14,15)>;

G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21), (1,10,18)(2,11,19)(3,12,20)(4,13,21)(5,14,15)(6,8,16)(7,9,17), (8,16)(9,17)(10,18)(11,19)(12,20)(13,21)(14,15) );

G=PermutationGroup([(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21)], [(1,10,18),(2,11,19),(3,12,20),(4,13,21),(5,14,15),(6,8,16),(7,9,17)], [(8,16),(9,17),(10,18),(11,19),(12,20),(13,21),(14,15)])

G:=TransitiveGroup(21,6);

Matrix representation of S3×C7 in GL2(𝔽29) generated by

160
016
,
022
2528
,
122
028
G:=sub<GL(2,GF(29))| [16,0,0,16],[0,25,22,28],[1,0,22,28] >;

S3×C7 in GAP, Magma, Sage, TeX

S_3\times C_7
% in TeX

G:=Group("S3xC7");
// GroupNames label

G:=SmallGroup(42,3);
// by ID

G=gap.SmallGroup(42,3);
# by ID

G:=PCGroup([3,-2,-7,-3,254]);
// Polycyclic

G:=Group<a,b,c|a^7=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

׿
×
𝔽