D7^2 - GroupNames

class	1	2A	2B	2C	7A	7B	7C	7D	7E	7F	7G	7H	7I	7J	7K	7L	7M	7N	7O	14A	14B	14C	14D	14E	14F
size	1	7	7	49	2	2	2	2	2	2	4	4	4	4	4	4	4	4	4	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	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	1	1	-1	linear of order 2
ρ₃	1	-1	-1	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
ρ₄	1	1	-1	-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
ρ₅	2	-2	0	0	2	2	2	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁴+ζ₇³	-ζ₇⁵-ζ₇²	-ζ₇⁴-ζ₇³	0	0	0	-ζ₇⁶-ζ₇	orthogonal lifted from D₁₄
ρ₆	2	-2	0	0	2	2	2	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁶+ζ₇	-ζ₇⁴-ζ₇³	-ζ₇⁶-ζ₇	0	0	0	-ζ₇⁵-ζ₇²	orthogonal lifted from D₁₄
ρ₇	2	-2	0	0	2	2	2	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁵+ζ₇²	-ζ₇⁶-ζ₇	-ζ₇⁵-ζ₇²	0	0	0	-ζ₇⁴-ζ₇³	orthogonal lifted from D₁₄
ρ₈	2	0	-2	0	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	2	2	2	ζ₇⁶+ζ₇	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁵+ζ₇²	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	0	0	-ζ₇⁴-ζ₇³	-ζ₇⁶-ζ₇	-ζ₇⁵-ζ₇²	0	orthogonal lifted from D₁₄
ρ₉	2	0	-2	0	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	2	2	2	ζ₇⁵+ζ₇²	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁴+ζ₇³	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	0	0	-ζ₇⁶-ζ₇	-ζ₇⁵-ζ₇²	-ζ₇⁴-ζ₇³	0	orthogonal lifted from D₁₄
ρ₁₀	2	2	0	0	2	2	2	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	0	0	0	ζ₇⁴+ζ₇³	orthogonal lifted from D₇
ρ₁₁	2	0	-2	0	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	2	2	2	ζ₇⁴+ζ₇³	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁶+ζ₇	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	0	0	-ζ₇⁵-ζ₇²	-ζ₇⁴-ζ₇³	-ζ₇⁶-ζ₇	0	orthogonal lifted from D₁₄
ρ₁₂	2	0	2	0	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	2	2	2	ζ₇⁵+ζ₇²	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁴+ζ₇³	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	0	0	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	0	orthogonal lifted from D₇
ρ₁₃	2	0	2	0	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	2	2	2	ζ₇⁴+ζ₇³	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁶+ζ₇	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	0	0	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	0	orthogonal lifted from D₇
ρ₁₄	2	0	2	0	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	2	2	2	ζ₇⁶+ζ₇	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁵+ζ₇²	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	0	0	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	0	orthogonal lifted from D₇
ρ₁₅	2	2	0	0	2	2	2	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	0	0	0	ζ₇⁶+ζ₇	orthogonal lifted from D₇
ρ₁₆	2	2	0	0	2	2	2	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁵+ζ₇²	ζ₇⁶+ζ₇	ζ₇⁵+ζ₇²	ζ₇⁴+ζ₇³	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	ζ₇⁶+ζ₇	ζ₇⁴+ζ₇³	ζ₇⁶+ζ₇	0	0	0	ζ₇⁵+ζ₇²	orthogonal lifted from D₇
ρ₁₇	4	0	0	0	2ζ₇⁶+2ζ₇	2ζ₇⁵+2ζ₇²	2ζ₇⁴+2ζ₇³	2ζ₇⁴+2ζ₇³	2ζ₇⁶+2ζ₇	2ζ₇⁵+2ζ₇²	-ζ₇⁵-ζ₇²-1	ζ₇⁴+ζ₇³+2	ζ₇⁵+ζ₇²+2	ζ₇⁶+ζ₇+2	-ζ₇⁶-ζ₇-1	-ζ₇⁴-ζ₇³-1	-ζ₇⁵-ζ₇²-1	-ζ₇⁶-ζ₇-1	-ζ₇⁴-ζ₇³-1	0	0	0	0	0	0	orthogonal faithful
ρ₁₈	4	0	0	0	2ζ₇⁴+2ζ₇³	2ζ₇⁶+2ζ₇	2ζ₇⁵+2ζ₇²	2ζ₇⁵+2ζ₇²	2ζ₇⁴+2ζ₇³	2ζ₇⁶+2ζ₇	-ζ₇⁶-ζ₇-1	ζ₇⁵+ζ₇²+2	ζ₇⁶+ζ₇+2	ζ₇⁴+ζ₇³+2	-ζ₇⁴-ζ₇³-1	-ζ₇⁵-ζ₇²-1	-ζ₇⁶-ζ₇-1	-ζ₇⁴-ζ₇³-1	-ζ₇⁵-ζ₇²-1	0	0	0	0	0	0	orthogonal faithful
ρ₁₉	4	0	0	0	2ζ₇⁴+2ζ₇³	2ζ₇⁶+2ζ₇	2ζ₇⁵+2ζ₇²	2ζ₇⁴+2ζ₇³	2ζ₇⁶+2ζ₇	2ζ₇⁵+2ζ₇²	ζ₇⁵+ζ₇²+2	-ζ₇⁵-ζ₇²-1	-ζ₇⁶-ζ₇-1	-ζ₇⁴-ζ₇³-1	-ζ₇⁵-ζ₇²-1	ζ₇⁴+ζ₇³+2	-ζ₇⁴-ζ₇³-1	ζ₇⁶+ζ₇+2	-ζ₇⁶-ζ₇-1	0	0	0	0	0	0	orthogonal faithful
ρ₂₀	4	0	0	0	2ζ₇⁴+2ζ₇³	2ζ₇⁶+2ζ₇	2ζ₇⁵+2ζ₇²	2ζ₇⁶+2ζ₇	2ζ₇⁵+2ζ₇²	2ζ₇⁴+2ζ₇³	-ζ₇⁵-ζ₇²-1	-ζ₇⁶-ζ₇-1	-ζ₇⁴-ζ₇³-1	-ζ₇⁵-ζ₇²-1	ζ₇⁴+ζ₇³+2	-ζ₇⁴-ζ₇³-1	ζ₇⁶+ζ₇+2	-ζ₇⁶-ζ₇-1	ζ₇⁵+ζ₇²+2	0	0	0	0	0	0	orthogonal faithful
ρ₂₁	4	0	0	0	2ζ₇⁶+2ζ₇	2ζ₇⁵+2ζ₇²	2ζ₇⁴+2ζ₇³	2ζ₇⁵+2ζ₇²	2ζ₇⁴+2ζ₇³	2ζ₇⁶+2ζ₇	-ζ₇⁴-ζ₇³-1	-ζ₇⁵-ζ₇²-1	-ζ₇⁶-ζ₇-1	-ζ₇⁴-ζ₇³-1	ζ₇⁶+ζ₇+2	-ζ₇⁶-ζ₇-1	ζ₇⁵+ζ₇²+2	-ζ₇⁵-ζ₇²-1	ζ₇⁴+ζ₇³+2	0	0	0	0	0	0	orthogonal faithful
ρ₂₂	4	0	0	0	2ζ₇⁵+2ζ₇²	2ζ₇⁴+2ζ₇³	2ζ₇⁶+2ζ₇	2ζ₇⁴+2ζ₇³	2ζ₇⁶+2ζ₇	2ζ₇⁵+2ζ₇²	-ζ₇⁶-ζ₇-1	-ζ₇⁴-ζ₇³-1	-ζ₇⁵-ζ₇²-1	-ζ₇⁶-ζ₇-1	ζ₇⁵+ζ₇²+2	-ζ₇⁵-ζ₇²-1	ζ₇⁴+ζ₇³+2	-ζ₇⁴-ζ₇³-1	ζ₇⁶+ζ₇+2	0	0	0	0	0	0	orthogonal faithful
ρ₂₃	4	0	0	0	2ζ₇⁵+2ζ₇²	2ζ₇⁴+2ζ₇³	2ζ₇⁶+2ζ₇	2ζ₇⁶+2ζ₇	2ζ₇⁵+2ζ₇²	2ζ₇⁴+2ζ₇³	-ζ₇⁴-ζ₇³-1	ζ₇⁶+ζ₇+2	ζ₇⁴+ζ₇³+2	ζ₇⁵+ζ₇²+2	-ζ₇⁵-ζ₇²-1	-ζ₇⁶-ζ₇-1	-ζ₇⁴-ζ₇³-1	-ζ₇⁵-ζ₇²-1	-ζ₇⁶-ζ₇-1	0	0	0	0	0	0	orthogonal faithful
ρ₂₄	4	0	0	0	2ζ₇⁵+2ζ₇²	2ζ₇⁴+2ζ₇³	2ζ₇⁶+2ζ₇	2ζ₇⁵+2ζ₇²	2ζ₇⁴+2ζ₇³	2ζ₇⁶+2ζ₇	ζ₇⁶+ζ₇+2	-ζ₇⁶-ζ₇-1	-ζ₇⁴-ζ₇³-1	-ζ₇⁵-ζ₇²-1	-ζ₇⁶-ζ₇-1	ζ₇⁵+ζ₇²+2	-ζ₇⁵-ζ₇²-1	ζ₇⁴+ζ₇³+2	-ζ₇⁴-ζ₇³-1	0	0	0	0	0	0	orthogonal faithful
ρ₂₅	4	0	0	0	2ζ₇⁶+2ζ₇	2ζ₇⁵+2ζ₇²	2ζ₇⁴+2ζ₇³	2ζ₇⁶+2ζ₇	2ζ₇⁵+2ζ₇²	2ζ₇⁴+2ζ₇³	ζ₇⁴+ζ₇³+2	-ζ₇⁴-ζ₇³-1	-ζ₇⁵-ζ₇²-1	-ζ₇⁶-ζ₇-1	-ζ₇⁴-ζ₇³-1	ζ₇⁶+ζ₇+2	-ζ₇⁶-ζ₇-1	ζ₇⁵+ζ₇²+2	-ζ₇⁵-ζ₇²-1	0	0	0	0	0	0	orthogonal faithful

Permutation representations of D₇²
►On 14 points - transitive group 14T13

Generators in S₁₄

(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)

(1 8)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)

(1 3 5 7 2 4 6)(8 13 11 9 14 12 10)

(1 11)(2 12)(3 13)(4 14)(5 8)(6 9)(7 10)

G:=sub<Sym(14)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,8)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9), (1,3,5,7,2,4,6)(8,13,11,9,14,12,10), (1,11)(2,12)(3,13)(4,14)(5,8)(6,9)(7,10)>;

G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14), (1,8)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9), (1,3,5,7,2,4,6)(8,13,11,9,14,12,10), (1,11)(2,12)(3,13)(4,14)(5,8)(6,9)(7,10) );

G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14)], [(1,8),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9)], [(1,3,5,7,2,4,6),(8,13,11,9,14,12,10)], [(1,11),(2,12),(3,13),(4,14),(5,8),(6,9),(7,10)]])

G:=TransitiveGroup(14,13);

►On 28 points - transitive group 28T36

Generators in S₂₈

(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)

(1 8)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(15 26)(16 25)(17 24)(18 23)(19 22)(20 28)(21 27)

(1 3 5 7 2 4 6)(8 13 11 9 14 12 10)(15 20 18 16 21 19 17)(22 24 26 28 23 25 27)

(1 21)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 27)(9 28)(10 22)(11 23)(12 24)(13 25)(14 26)

G:=sub<Sym(28)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28), (1,8)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(15,26)(16,25)(17,24)(18,23)(19,22)(20,28)(21,27), (1,3,5,7,2,4,6)(8,13,11,9,14,12,10)(15,20,18,16,21,19,17)(22,24,26,28,23,25,27), (1,21)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26)>;

G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28), (1,8)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(15,26)(16,25)(17,24)(18,23)(19,22)(20,28)(21,27), (1,3,5,7,2,4,6)(8,13,11,9,14,12,10)(15,20,18,16,21,19,17)(22,24,26,28,23,25,27), (1,21)(2,15)(3,16)(4,17)(5,18)(6,19)(7,20)(8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26) );

G=PermutationGroup([[(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28)], [(1,8),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(15,26),(16,25),(17,24),(18,23),(19,22),(20,28),(21,27)], [(1,3,5,7,2,4,6),(8,13,11,9,14,12,10),(15,20,18,16,21,19,17),(22,24,26,28,23,25,27)], [(1,21),(2,15),(3,16),(4,17),(5,18),(6,19),(7,20),(8,27),(9,28),(10,22),(11,23),(12,24),(13,25),(14,26)]])

G:=TransitiveGroup(28,36);

D₇² is a maximal subgroup of D₇≀C₂
D₇² is a maximal quotient of Dic₇⋊₂D₇ C₇²⋊₂D₄ C₇⋊D₂₈ C₇²⋊₂Q₈

Polynomial with Galois group D₇² over ℚ

action	f(x)	Disc(f)
14T13	x¹⁴-2x¹³-29x¹²-222x¹¹-352x¹⁰+3498x⁹+18163x⁸+46467x⁷+92188x⁶+128405x⁵+96637x⁴+31142x³+7064x²+6304x+2432	2³⁶·7⁴·23²·71¹¹·151⁷·1423²·1723²·161039²·1768517²

Matrix representation of D₇² ►in GL₄(𝔽₂₉) generated by

1	0	0	0
0	1	0	0
0	0	7	4
0	0	7	0

28	0	0	0
0	28	0	0
0	0	19	11
0	0	20	10

10	18	0	0
17	22	0	0
0	0	1	0
0	0	0	1

28	1	0	0
0	1	0	0
0	0	28	0
0	0	0	28

G:=sub<GL(4,GF(29))| [1,0,0,0,0,1,0,0,0,0,7,7,0,0,4,0],[28,0,0,0,0,28,0,0,0,0,19,20,0,0,11,10],[10,17,0,0,18,22,0,0,0,0,1,0,0,0,0,1],[28,0,0,0,1,1,0,0,0,0,28,0,0,0,0,28] >;

D₇² in GAP, Magma, Sage, TeX

D_7^2

% in TeX

G:=Group("D7^2");

// GroupNames label

G:=SmallGroup(196,9);

// by ID

G=gap.SmallGroup(196,9);

# by ID

G:=PCGroup([4,-2,-2,-7,-7,150,2691]);

// Polycyclic

G:=Group<a,b,c,d|a^7=b^2=c^7=d^2=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=c^-1>;

// generators/relations

Export

Subgroup lattice of D₇² in TeX
Character table of D₇² in TeX

G = D72 order 196 = 22·72

Direct product of D7 and D7

G = D₇² order 196 = 2²·7²

Direct product of D₇ and D₇