C7:5F7 - GroupNames

class	1	2	3A	3B	6A	6B	7A	7B	7C	7D	7E	7F	7G	7H	7I	7J	21A	21B	21C	21D	21E	21F
size	1	49	7	7	49	49	2	2	2	6	6	6	6	6	6	6	14	14	14	14	14	14
ρ₁	1	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	1	linear of order 2
ρ₃	1	1	ζ₃²	ζ₃	ζ₃	ζ₃²	1	1	1	1	1	1	1	1	1	1	ζ₃	ζ₃	ζ₃	ζ₃²	ζ₃²	ζ₃²	linear of order 3
ρ₄	1	1	ζ₃	ζ₃²	ζ₃²	ζ₃	1	1	1	1	1	1	1	1	1	1	ζ₃²	ζ₃²	ζ₃²	ζ₃	ζ₃	ζ₃	linear of order 3
ρ₅	1	-1	ζ₃	ζ₃²	ζ₆	ζ₆⁵	1	1	1	1	1	1	1	1	1	1	ζ₃²	ζ₃²	ζ₃²	ζ₃	ζ₃	ζ₃	linear of order 6
ρ₆	1	-1	ζ₃²	ζ₃	ζ₆⁵	ζ₆	1	1	1	1	1	1	1	1	1	1	ζ₃	ζ₃	ζ₃	ζ₃²	ζ₃²	ζ₃²	linear of order 6
ρ₇	2	0	2	2	0	0	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	2	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	orthogonal lifted from D₇
ρ₈	2	0	2	2	0	0	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	2	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	orthogonal lifted from D₇
ρ₉	2	0	2	2	0	0	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	2	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	orthogonal lifted from D₇
ρ₁₀	2	0	-1-√-3	-1+√-3	0	0	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	2	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₃ζ₇⁴+ζ₃ζ₇³	ζ₃ζ₇⁶+ζ₃ζ₇	ζ₃ζ₇⁵+ζ₃ζ₇²	ζ₃²ζ₇⁶+ζ₃²ζ₇	ζ₃²ζ₇⁵+ζ₃²ζ₇²	ζ₃²ζ₇⁴+ζ₃²ζ₇³	complex lifted from C₃×D₇
ρ₁₁	2	0	-1+√-3	-1-√-3	0	0	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	2	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₃²ζ₇⁴+ζ₃²ζ₇³	ζ₃²ζ₇⁶+ζ₃²ζ₇	ζ₃²ζ₇⁵+ζ₃²ζ₇²	ζ₃ζ₇⁶+ζ₃ζ₇	ζ₃ζ₇⁵+ζ₃ζ₇²	ζ₃ζ₇⁴+ζ₃ζ₇³	complex lifted from C₃×D₇
ρ₁₂	2	0	-1+√-3	-1-√-3	0	0	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	2	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₃²ζ₇⁶+ζ₃²ζ₇	ζ₃²ζ₇⁵+ζ₃²ζ₇²	ζ₃²ζ₇⁴+ζ₃²ζ₇³	ζ₃ζ₇⁵+ζ₃ζ₇²	ζ₃ζ₇⁴+ζ₃ζ₇³	ζ₃ζ₇⁶+ζ₃ζ₇	complex lifted from C₃×D₇
ρ₁₃	2	0	-1-√-3	-1+√-3	0	0	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	2	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₃ζ₇⁵+ζ₃ζ₇²	ζ₃ζ₇⁴+ζ₃ζ₇³	ζ₃ζ₇⁶+ζ₃ζ₇	ζ₃²ζ₇⁴+ζ₃²ζ₇³	ζ₃²ζ₇⁶+ζ₃²ζ₇	ζ₃²ζ₇⁵+ζ₃²ζ₇²	complex lifted from C₃×D₇
ρ₁₄	2	0	-1-√-3	-1+√-3	0	0	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	2	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₃ζ₇⁶+ζ₃ζ₇	ζ₃ζ₇⁵+ζ₃ζ₇²	ζ₃ζ₇⁴+ζ₃ζ₇³	ζ₃²ζ₇⁵+ζ₃²ζ₇²	ζ₃²ζ₇⁴+ζ₃²ζ₇³	ζ₃²ζ₇⁶+ζ₃²ζ₇	complex lifted from C₃×D₇
ρ₁₅	2	0	-1+√-3	-1-√-3	0	0	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	2	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₃²ζ₇⁵+ζ₃²ζ₇²	ζ₃²ζ₇⁴+ζ₃²ζ₇³	ζ₃²ζ₇⁶+ζ₃²ζ₇	ζ₃ζ₇⁴+ζ₃ζ₇³	ζ₃ζ₇⁶+ζ₃ζ₇	ζ₃ζ₇⁵+ζ₃ζ₇²	complex lifted from C₃×D₇
ρ₁₆	6	0	0	0	0	0	6	6	6	-1	-1	-1	-1	-1	-1	-1	0	0	0	0	0	0	orthogonal lifted from F₇
ρ₁₇	6	0	0	0	0	0	3ζ₇⁵+3ζ₇²	3ζ₇⁴+3ζ₇³	3ζ₇⁶+3ζ₇	-ζ₇⁶-ζ₇+1	ζ₇⁶+2ζ₇⁴+2ζ₇³+ζ₇	-1	-ζ₇⁵-ζ₇²+1	2ζ₇⁶+ζ₇⁵+ζ₇²+2ζ₇	2ζ₇⁵+ζ₇⁴+ζ₇³+2ζ₇²	-ζ₇⁴-ζ₇³+1	0	0	0	0	0	0	orthogonal faithful
ρ₁₈	6	0	0	0	0	0	3ζ₇⁵+3ζ₇²	3ζ₇⁴+3ζ₇³	3ζ₇⁶+3ζ₇	2ζ₇⁶+ζ₇⁵+ζ₇²+2ζ₇	-ζ₇⁴-ζ₇³+1	-1	2ζ₇⁵+ζ₇⁴+ζ₇³+2ζ₇²	-ζ₇⁶-ζ₇+1	-ζ₇⁵-ζ₇²+1	ζ₇⁶+2ζ₇⁴+2ζ₇³+ζ₇	0	0	0	0	0	0	orthogonal faithful
ρ₁₉	6	0	0	0	0	0	3ζ₇⁴+3ζ₇³	3ζ₇⁶+3ζ₇	3ζ₇⁵+3ζ₇²	2ζ₇⁵+ζ₇⁴+ζ₇³+2ζ₇²	-ζ₇⁶-ζ₇+1	-1	ζ₇⁶+2ζ₇⁴+2ζ₇³+ζ₇	-ζ₇⁵-ζ₇²+1	-ζ₇⁴-ζ₇³+1	2ζ₇⁶+ζ₇⁵+ζ₇²+2ζ₇	0	0	0	0	0	0	orthogonal faithful
ρ₂₀	6	0	0	0	0	0	3ζ₇⁴+3ζ₇³	3ζ₇⁶+3ζ₇	3ζ₇⁵+3ζ₇²	-ζ₇⁵-ζ₇²+1	2ζ₇⁶+ζ₇⁵+ζ₇²+2ζ₇	-1	-ζ₇⁴-ζ₇³+1	2ζ₇⁵+ζ₇⁴+ζ₇³+2ζ₇²	ζ₇⁶+2ζ₇⁴+2ζ₇³+ζ₇	-ζ₇⁶-ζ₇+1	0	0	0	0	0	0	orthogonal faithful
ρ₂₁	6	0	0	0	0	0	3ζ₇⁶+3ζ₇	3ζ₇⁵+3ζ₇²	3ζ₇⁴+3ζ₇³	-ζ₇⁴-ζ₇³+1	2ζ₇⁵+ζ₇⁴+ζ₇³+2ζ₇²	-1	-ζ₇⁶-ζ₇+1	ζ₇⁶+2ζ₇⁴+2ζ₇³+ζ₇	2ζ₇⁶+ζ₇⁵+ζ₇²+2ζ₇	-ζ₇⁵-ζ₇²+1	0	0	0	0	0	0	orthogonal faithful
ρ₂₂	6	0	0	0	0	0	3ζ₇⁶+3ζ₇	3ζ₇⁵+3ζ₇²	3ζ₇⁴+3ζ₇³	ζ₇⁶+2ζ₇⁴+2ζ₇³+ζ₇	-ζ₇⁵-ζ₇²+1	-1	2ζ₇⁶+ζ₇⁵+ζ₇²+2ζ₇	-ζ₇⁴-ζ₇³+1	-ζ₇⁶-ζ₇+1	2ζ₇⁵+ζ₇⁴+ζ₇³+2ζ₇²	0	0	0	0	0	0	orthogonal faithful

Permutation representations of C₇⋊₅F₇
►On 21 points - transitive group 21T16

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

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

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

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

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

G:=TransitiveGroup(21,16);

Matrix representation of C₇⋊₅F₇ ►in GL₆(𝔽₄₃)

23	23	0	0	0	0
20	35	0	0	0	0
0	0	23	23	0	0
0	0	20	35	0	0
0	0	0	0	23	23
0	0	0	0	20	35

8	42	0	0	0	0
1	0	0	0	0	0
0	0	20	35	0	0
0	0	8	42	0	0
0	0	0	0	35	20
0	0	0	0	23	23

0	0	0	0	1	0
0	0	0	0	8	42
1	0	0	0	0	0
8	42	0	0	0	0
0	0	1	0	0	0
0	0	8	42	0	0

G:=sub<GL(6,GF(43))| [23,20,0,0,0,0,23,35,0,0,0,0,0,0,23,20,0,0,0,0,23,35,0,0,0,0,0,0,23,20,0,0,0,0,23,35],[8,1,0,0,0,0,42,0,0,0,0,0,0,0,20,8,0,0,0,0,35,42,0,0,0,0,0,0,35,23,0,0,0,0,20,23],[0,0,1,8,0,0,0,0,0,42,0,0,0,0,0,0,1,8,0,0,0,0,0,42,1,8,0,0,0,0,0,42,0,0,0,0] >;

C₇⋊₅F₇ in GAP, Magma, Sage, TeX

C_7\rtimes_5F_7

% in TeX

G:=Group("C7:5F7");

// GroupNames label

G:=SmallGroup(294,10);

// by ID

G=gap.SmallGroup(294,10);

# by ID

G:=PCGroup([4,-2,-3,-7,-7,434,4035,679]);

// Polycyclic

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

// generators/relations

Export

Subgroup lattice of C₇⋊₅F₇ in TeX
Character table of C₇⋊₅F₇ in TeX

G = C7⋊5F7 order 294 = 2·3·72

The semidirect product of C7 and F7 acting via F7/C7⋊C3=C2

G = C₇⋊₅F₇ order 294 = 2·3·7²

The semidirect product of C₇ and F₇ acting via F₇/C₇⋊C₃=C₂