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G = S3×C7⋊C3  order 126 = 2·32·7

Direct product of S3 and C7⋊C3

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: S3×C7⋊C3, C21⋊3C6, (S3×C7)⋊C3, C7⋊2(C3×S3), C3⋊(C2×C7⋊C3), (C3×C7⋊C3)⋊3C2, SmallGroup(126,8)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C21 — S3×C7⋊C3
Chief series
C1 — C7 — C21 — C3×C7⋊C3 — S3×C7⋊C3
Lower central
C21 — S3×C7⋊C3
Upper central
C1

Generators and relations for S3×C7⋊C3
 G = < a,b,c,d | a3=b2=c7=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >

C1
3C2
C3
7C3
14C3
C7
S3
21C6
7C32
3C14
C21
C7⋊C3
2C7⋊C3
7C3×S3
S3×C7
3C2×C7⋊C3
C3×C7⋊C3
S3×C7⋊C3

Character table of S3×C7⋊C3

 class 123A3B3C3D3E6A6B7A7B14A14B21A21B
 size 1327714142121339966
ρ1111111111111111    trivial
ρ21-111111-1-111-1-111    linear of order 2
ρ3111ζ3ζ32ζ3ζ32ζ32ζ3111111    linear of order 3
ρ41-11ζ3ζ32ζ3ζ32ζ6ζ6511-1-111    linear of order 6
ρ51-11ζ32ζ3ζ32ζ3ζ65ζ611-1-111    linear of order 6
ρ6111ζ32ζ3ζ32ζ3ζ3ζ32111111    linear of order 3
ρ720-122-1-1002200-1-1    orthogonal lifted from S3
ρ820-1-1+√-3-1-√-3ζ65ζ6002200-1-1    complex lifted from C3×S3
ρ920-1-1-√-3-1+√-3ζ6ζ65002200-1-1    complex lifted from C3×S3
ρ10333000000-1+√-7/2-1-√-7/2-1+√-7/2-1-√-7/2-1-√-7/2-1+√-7/2    complex lifted from C7⋊C3
ρ113-33000000-1+√-7/2-1-√-7/21-√-7/21+√-7/2-1-√-7/2-1+√-7/2    complex lifted from C2×C7⋊C3
ρ12333000000-1-√-7/2-1+√-7/2-1-√-7/2-1+√-7/2-1+√-7/2-1-√-7/2    complex lifted from C7⋊C3
ρ133-33000000-1-√-7/2-1+√-7/21+√-7/21-√-7/2-1+√-7/2-1-√-7/2    complex lifted from C2×C7⋊C3
ρ1460-3000000-1-√-7-1+√-7001-√-7/21+√-7/2    complex faithful
ρ1560-3000000-1+√-7-1-√-7001+√-7/21-√-7/2    complex faithful

Permutation representations of S3×C7⋊C3
►On 21 points - transitive group 21T11
Generators in S21
(1 8 15)(2 9 16)(3 10 17)(4 11 18)(5 12 19)(6 13 20)(7 14 21)

(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)

(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)

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

S3×C7⋊C3 is a maximal subgroup of   C63⋊C6  C63⋊6C6  C7⋊He3⋊C2
S3×C7⋊C3 is a maximal quotient of   C63⋊C6  C63⋊6C6  C7⋊He3⋊C2

Matrix representation of S3×C7⋊C3 ►in GL5(𝔽43)

4242000
10000
00100
00010
00001
,
10000
4242000
00100
00010
00001
,
10000
01000
00424218
00101
000125
,
360000
036000
000191
001125
0002542

G:=sub<GL(5,GF(43))| [42,1,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,42,0,0,0,0,42,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,42,1,0,0,0,42,0,1,0,0,18,1,25],[36,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,19,1,25,0,0,1,25,42] >;
 

S3×C7⋊C3 in GAP, Magma, Sage, TeX 

S_3\times C_7\rtimes C_3
 
% in TeX
 
G:=Group("S3xC7:C3");
 
// GroupNames label
 
G:=SmallGroup(126,8);
 
// by ID
 
G=gap.SmallGroup(126,8);
 
# by ID
 
G:=PCGroup([4,-2,-3,-3,-7,146,295]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^3=b^2=c^7=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
 
// generators/relations
 

Export

Subgroup lattice of S3×C7⋊C3 in TeX
Character table of S3×C7⋊C3 in TeX

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