Copied to
clipboard

## G = C3×C7⋊C3order 63 = 32·7

### Direct product of C3 and C7⋊C3

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 >

Character table of C3×C7⋊C3

 class 1 3A 3B 3C 3D 3E 3F 3G 3H 7A 7B 21A 21B 21C 21D size 1 1 1 7 7 7 7 7 7 3 3 3 3 3 3 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 linear of order 3 ρ3 1 ζ3 ζ32 1 1 ζ3 ζ3 ζ32 ζ32 1 1 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ4 1 ζ3 ζ32 ζ32 ζ3 ζ32 1 ζ3 1 1 1 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ5 1 ζ32 ζ3 ζ32 ζ3 1 ζ3 1 ζ32 1 1 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ6 1 ζ32 ζ3 ζ3 ζ32 ζ3 1 ζ32 1 1 1 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ7 1 1 1 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 linear of order 3 ρ8 1 ζ32 ζ3 1 1 ζ32 ζ32 ζ3 ζ3 1 1 ζ32 ζ32 ζ3 ζ3 linear of order 3 ρ9 1 ζ3 ζ32 ζ3 ζ32 1 ζ32 1 ζ3 1 1 ζ3 ζ3 ζ32 ζ32 linear of order 3 ρ10 3 3 3 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 complex lifted from C7⋊C3 ρ11 3 3 3 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 -1-√-7/2 -1+√-7/2 -1+√-7/2 -1-√-7/2 complex lifted from C7⋊C3 ρ12 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 complex faithful ρ13 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ32ζ76+ζ32ζ75+ζ32ζ73 complex faithful ρ14 3 -3+3√-3/2 -3-3√-3/2 0 0 0 0 0 0 -1-√-7/2 -1+√-7/2 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ76+ζ3ζ75+ζ3ζ73 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 complex faithful ρ15 3 -3-3√-3/2 -3+3√-3/2 0 0 0 0 0 0 -1+√-7/2 -1-√-7/2 ζ32ζ76+ζ32ζ75+ζ32ζ73 ζ32ζ74+ζ32ζ72+ζ32ζ7 ζ3ζ74+ζ3ζ72+ζ3ζ7 ζ3ζ76+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  C633C3  C7⋊He3
C3×C7⋊C3 is a maximal quotient of   C63⋊C3  C633C3  C21.C32  C7⋊He3

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

 6 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 18 19 1 0 1 0 0 0 0 1 0
,
 6 0 0 0 0 1 0 0 0 24 42 42 0 0 1 0
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

׿
×
𝔽