Copied to
clipboard

## G = C3×F7order 126 = 2·32·7

### Direct product of C3 and F7

Aliases: C3×F7, D7⋊C32, C212C6, C7⋊C3⋊C6, C7⋊(C3×C6), (C3×D7)⋊C3, (C3×C7⋊C3)⋊2C2, SmallGroup(126,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C3×F7
 Chief series C1 — C7 — C21 — C3×C7⋊C3 — C3×F7
 Lower central C7 — C3×F7
 Upper central C1 — C3

Generators and relations for C3×F7
G = < a,b,c | a3=b7=c6=1, ab=ba, ac=ca, cbc-1=b5 >

Character table of C3×F7

 class 1 2 3A 3B 3C 3D 3E 3F 3G 3H 6A 6B 6C 6D 6E 6F 6G 6H 7 21A 21B size 1 7 1 1 7 7 7 7 7 7 7 7 7 7 7 7 7 7 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 linear of order 2 ρ3 1 -1 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ32 ζ65 ζ65 ζ6 ζ65 ζ6 -1 ζ6 -1 1 ζ32 ζ3 linear of order 6 ρ4 1 1 ζ32 ζ3 1 1 ζ32 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 ζ3 ζ32 1 ζ32 1 1 ζ32 ζ3 linear of order 3 ρ5 1 -1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 1 -1 ζ65 ζ6 ζ6 -1 ζ65 ζ65 ζ6 1 ζ32 ζ3 linear of order 6 ρ6 1 -1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ65 -1 -1 ζ6 ζ6 ζ6 ζ65 ζ65 1 1 1 linear of order 6 ρ7 1 -1 ζ32 ζ3 ζ3 ζ32 1 1 ζ32 ζ3 ζ6 ζ65 ζ6 -1 ζ65 ζ6 -1 ζ65 1 ζ32 ζ3 linear of order 6 ρ8 1 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 1 ζ32 ζ32 ζ32 ζ3 ζ3 1 1 1 linear of order 3 ρ9 1 1 ζ3 ζ32 ζ32 ζ3 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 1 ζ32 ζ3 1 ζ32 1 ζ3 ζ32 linear of order 3 ρ10 1 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 1 ζ3 ζ3 ζ3 ζ32 ζ32 1 1 1 linear of order 3 ρ11 1 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 1 1 1 ζ3 ζ32 ζ32 1 ζ3 ζ3 ζ32 1 ζ32 ζ3 linear of order 3 ρ12 1 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 1 1 ζ32 ζ3 ζ3 1 ζ32 ζ32 ζ3 1 ζ3 ζ32 linear of order 3 ρ13 1 1 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 ζ32 ζ3 1 ζ3 1 1 ζ3 ζ32 linear of order 3 ρ14 1 -1 ζ3 ζ32 1 1 ζ3 ζ32 ζ32 ζ3 ζ6 ζ6 ζ65 ζ6 ζ65 -1 ζ65 -1 1 ζ3 ζ32 linear of order 6 ρ15 1 -1 ζ3 ζ32 ζ32 ζ3 1 1 ζ3 ζ32 ζ65 ζ6 ζ65 -1 ζ6 ζ65 -1 ζ6 1 ζ3 ζ32 linear of order 6 ρ16 1 -1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 1 1 -1 ζ6 ζ65 ζ65 -1 ζ6 ζ6 ζ65 1 ζ3 ζ32 linear of order 6 ρ17 1 -1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 ζ6 -1 -1 ζ65 ζ65 ζ65 ζ6 ζ6 1 1 1 linear of order 6 ρ18 1 1 ζ32 ζ3 ζ3 ζ32 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 1 ζ3 ζ32 1 ζ3 1 ζ32 ζ3 linear of order 3 ρ19 6 0 6 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from F7 ρ20 6 0 -3+3√-3 -3-3√-3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 ζ65 ζ6 complex faithful ρ21 6 0 -3-3√-3 -3+3√-3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 ζ6 ζ65 complex faithful

Permutation representations of C3×F7
On 21 points - transitive group 21T9
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 15 8)(2 18 10 7 19 13)(3 21 12 6 16 11)(4 17 14 5 20 9)

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

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

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

G:=TransitiveGroup(21,9);

C3×F7 is a maximal subgroup of   C93F7  C94F7  D7⋊He3
C3×F7 is a maximal quotient of   C93F7  C94F7  C32.F7  D7⋊He3

Matrix representation of C3×F7 in GL7(𝔽43)

 6 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 42 0 1 0 0 0 0 42 0 0 1 0 0 0 42 0 0 0 1 0 0 42 0 0 0 0 1 0 42 0 0 0 0 0 1 42
,
 42 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0

G:=sub<GL(7,GF(43))| [6,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,42,42,42,42,42,42],[42,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0] >;

C3×F7 in GAP, Magma, Sage, TeX

C_3\times F_7
% in TeX

G:=Group("C3xF7");
// GroupNames label

G:=SmallGroup(126,7);
// by ID

G=gap.SmallGroup(126,7);
# by ID

G:=PCGroup([4,-2,-3,-3,-7,1731,583]);
// Polycyclic

G:=Group<a,b,c|a^3=b^7=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

Export

׿
×
𝔽