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G = C3×C7⋊C3  order 63 = 32·7

Direct product of C3 and C7⋊C3

direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary

Aliases: C3×C7⋊C3, C21⋊C3, C7⋊C32, SmallGroup(63,3)

Series: Derived ►Chief ►Lower central ►Upper central

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

Generators and relations for C3×C7⋊C3
 G = < a,b,c | a3=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

C1
C3
7C3
7C3
7C3
C7
7C32
C7⋊C3
C7⋊C3
C7⋊C3
C21
C3×C7⋊C3

Character table of C3×C7⋊C3

 class 13A3B3C3D3E3F3G3H7A7B21A21B21C21D
 size 111777777333333
ρ1111111111111111    trivial
ρ2111ζ3ζ32ζ32ζ3ζ3ζ32111111    linear of order 3
ρ31ζ3ζ3211ζ3ζ3ζ32ζ3211ζ3ζ3ζ32ζ32    linear of order 3
ρ41ζ3ζ32ζ32ζ3ζ321ζ3111ζ3ζ3ζ32ζ32    linear of order 3
ρ51ζ32ζ3ζ32ζ31ζ31ζ3211ζ32ζ32ζ3ζ3    linear of order 3
ρ61ζ32ζ3ζ3ζ32ζ31ζ32111ζ32ζ32ζ3ζ3    linear of order 3
ρ7111ζ32ζ3ζ3ζ32ζ32ζ3111111    linear of order 3
ρ81ζ32ζ311ζ32ζ32ζ3ζ311ζ32ζ32ζ3ζ3    linear of order 3
ρ91ζ3ζ32ζ3ζ321ζ321ζ311ζ3ζ3ζ32ζ32    linear of order 3
ρ10333000000-1-√-7/2-1+√-7/2-1+√-7/2-1-√-7/2-1-√-7/2-1+√-7/2    complex lifted from C7⋊C3
ρ11333000000-1+√-7/2-1-√-7/2-1-√-7/2-1+√-7/2-1+√-7/2-1-√-7/2    complex lifted from C7⋊C3
ρ123-3-3√-3/2-3+3√-3/2000000-1-√-7/2-1+√-7/2ζ32ζ74+ζ32ζ72+ζ32ζ7ζ32ζ76+ζ32ζ75+ζ32ζ73ζ3ζ76+ζ3ζ75+ζ3ζ73ζ3ζ74+ζ3ζ72+ζ3ζ7    complex faithful
ρ133-3+3√-3/2-3-3√-3/2000000-1+√-7/2-1-√-7/2ζ3ζ76+ζ3ζ75+ζ3ζ73ζ3ζ74+ζ3ζ72+ζ3ζ7ζ32ζ74+ζ32ζ72+ζ32ζ7ζ32ζ76+ζ32ζ75+ζ32ζ73    complex faithful
ρ143-3+3√-3/2-3-3√-3/2000000-1-√-7/2-1+√-7/2ζ3ζ74+ζ3ζ72+ζ3ζ7ζ3ζ76+ζ3ζ75+ζ3ζ73ζ32ζ76+ζ32ζ75+ζ32ζ73ζ32ζ74+ζ32ζ72+ζ32ζ7    complex faithful
ρ153-3-3√-3/2-3+3√-3/2000000-1+√-7/2-1-√-7/2ζ32ζ76+ζ32ζ75+ζ32ζ73ζ32ζ74+ζ32ζ72+ζ32ζ7ζ3ζ74+ζ3ζ72+ζ3ζ7ζ3ζ76+ζ3ζ75+ζ3ζ73    complex faithful

Permutation representations of C3×C7⋊C3
►On 21 points - transitive group 21T7
Generators in S21
(1 15 8)(2 16 9)(3 17 10)(4 18 11)(5 19 12)(6 20 13)(7 21 14)

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

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

C3×C7⋊C3 is a maximal subgroup of   C3⋊F7  C63⋊C3  C63⋊3C3  C7⋊He3
C3×C7⋊C3 is a maximal quotient of   C63⋊C3  C63⋊3C3  C21.C32  C7⋊He3

Matrix representation of C3×C7⋊C3 ►in GL4(𝔽43) generated by

6000
0100
0010
0001
,
1000
018191
0100
0010
,
6000
0100
0244242
0010
G:=sub<GL(4,GF(43))| [6,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,18,1,0,0,19,0,1,0,1,0,0],[6,0,0,0,0,1,24,0,0,0,42,1,0,0,42,0] >;
 

C3×C7⋊C3 in GAP, Magma, Sage, TeX 

C_3\times C_7\rtimes C_3
 
% in TeX
 
G:=Group("C3xC7:C3");
 
// GroupNames label
 
G:=SmallGroup(63,3);
 
// by ID
 
G=gap.SmallGroup(63,3);
 
# by ID
 
G:=PCGroup([3,-3,-3,-7,164]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
 
// generators/relations
 

Export

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

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