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G = C6×D7order 84 = 22·3·7

Direct product of C6 and D7

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C6×D7, C422C2, C143C6, C213C22, C73(C2×C6), SmallGroup(84,12)

Series: Derived Chief Lower central Upper central

C1C7 — C6×D7
C1C7C21C3×D7 — C6×D7
C7 — C6×D7
C1C6

Generators and relations for C6×D7
 G = < a,b,c | a6=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

7C2
7C2
7C22
7C6
7C6
7C2×C6

Character table of C6×D7

 class 12A2B2C3A3B6A6B6C6D6E6F7A7B7C14A14B14C21A21B21C21D21E21F42A42B42C42D42E42F
 size 117711117777222222222222222222
ρ1111111111111111111111111111111    trivial
ρ21-1-1111-1-1-11-11111-1-1-1111111-1-1-1-1-1-1    linear of order 2
ρ31-11-111-1-11-11-1111-1-1-1111111-1-1-1-1-1-1    linear of order 2
ρ411-1-11111-1-1-1-1111111111111111111    linear of order 2
ρ51111ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ32111111ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 3
ρ61-1-11ζ32ζ3ζ65ζ6ζ6ζ3ζ65ζ32111-1-1-1ζ32ζ3ζ3ζ32ζ32ζ3ζ65ζ65ζ65ζ6ζ6ζ6    linear of order 6
ρ711-1-1ζ32ζ3ζ3ζ32ζ6ζ65ζ65ζ6111111ζ32ζ3ζ3ζ32ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ32    linear of order 6
ρ81-11-1ζ32ζ3ζ65ζ6ζ32ζ65ζ3ζ6111-1-1-1ζ32ζ3ζ3ζ32ζ32ζ3ζ65ζ65ζ65ζ6ζ6ζ6    linear of order 6
ρ91-11-1ζ3ζ32ζ6ζ65ζ3ζ6ζ32ζ65111-1-1-1ζ3ζ32ζ32ζ3ζ3ζ32ζ6ζ6ζ6ζ65ζ65ζ65    linear of order 6
ρ1011-1-1ζ3ζ32ζ32ζ3ζ65ζ6ζ6ζ65111111ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 6
ρ111-1-11ζ3ζ32ζ6ζ65ζ65ζ32ζ6ζ3111-1-1-1ζ3ζ32ζ32ζ3ζ3ζ32ζ6ζ6ζ6ζ65ζ65ζ65    linear of order 6
ρ121111ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ3111111ζ3ζ32ζ32ζ3ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ3    linear of order 3
ρ132-20022-2-20000ζ767ζ7572ζ747376775727473ζ767ζ767ζ7572ζ7572ζ7473ζ74737473767757275727473767    orthogonal lifted from D14
ρ142-20022-2-20000ζ7473ζ767ζ757274737677572ζ7473ζ7473ζ767ζ767ζ7572ζ75727572747376776775727473    orthogonal lifted from D14
ρ15220022220000ζ7473ζ767ζ7572ζ7473ζ767ζ7572ζ7473ζ7473ζ767ζ767ζ7572ζ7572ζ7572ζ7473ζ767ζ767ζ7572ζ7473    orthogonal lifted from D7
ρ162-20022-2-20000ζ7572ζ7473ζ76775727473767ζ7572ζ7572ζ7473ζ7473ζ767ζ7677677572747374737677572    orthogonal lifted from D14
ρ17220022220000ζ7572ζ7473ζ767ζ7572ζ7473ζ767ζ7572ζ7572ζ7473ζ7473ζ767ζ767ζ767ζ7572ζ7473ζ7473ζ767ζ7572    orthogonal lifted from D7
ρ18220022220000ζ767ζ7572ζ7473ζ767ζ7572ζ7473ζ767ζ767ζ7572ζ7572ζ7473ζ7473ζ7473ζ767ζ7572ζ7572ζ7473ζ767    orthogonal lifted from D7
ρ192200-1--3-1+-3-1+-3-1--30000ζ7572ζ7473ζ767ζ7572ζ7473ζ767ζ32ζ7532ζ72ζ3ζ753ζ72ζ3ζ743ζ73ζ32ζ7432ζ73ζ32ζ7632ζ7ζ3ζ763ζ7ζ3ζ763ζ7ζ3ζ753ζ72ζ3ζ743ζ73ζ32ζ7432ζ73ζ32ζ7632ζ7ζ32ζ7532ζ72    complex lifted from C3×D7
ρ202-200-1--3-1+-31--31+-30000ζ7572ζ7473ζ76775727473767ζ32ζ7532ζ72ζ3ζ753ζ72ζ3ζ743ζ73ζ32ζ7432ζ73ζ32ζ7632ζ7ζ3ζ763ζ73ζ763ζ73ζ753ζ723ζ743ζ7332ζ7432ζ7332ζ7632ζ732ζ7532ζ72    complex faithful
ρ212-200-1--3-1+-31--31+-30000ζ7473ζ767ζ757274737677572ζ32ζ7432ζ73ζ3ζ743ζ73ζ3ζ763ζ7ζ32ζ7632ζ7ζ32ζ7532ζ72ζ3ζ753ζ723ζ753ζ723ζ743ζ733ζ763ζ732ζ7632ζ732ζ7532ζ7232ζ7432ζ73    complex faithful
ρ222200-1+-3-1--3-1--3-1+-30000ζ7473ζ767ζ7572ζ7473ζ767ζ7572ζ3ζ743ζ73ζ32ζ7432ζ73ζ32ζ7632ζ7ζ3ζ763ζ7ζ3ζ753ζ72ζ32ζ7532ζ72ζ32ζ7532ζ72ζ32ζ7432ζ73ζ32ζ7632ζ7ζ3ζ763ζ7ζ3ζ753ζ72ζ3ζ743ζ73    complex lifted from C3×D7
ρ232200-1--3-1+-3-1+-3-1--30000ζ7473ζ767ζ7572ζ7473ζ767ζ7572ζ32ζ7432ζ73ζ3ζ743ζ73ζ3ζ763ζ7ζ32ζ7632ζ7ζ32ζ7532ζ72ζ3ζ753ζ72ζ3ζ753ζ72ζ3ζ743ζ73ζ3ζ763ζ7ζ32ζ7632ζ7ζ32ζ7532ζ72ζ32ζ7432ζ73    complex lifted from C3×D7
ρ242200-1+-3-1--3-1--3-1+-30000ζ767ζ7572ζ7473ζ767ζ7572ζ7473ζ3ζ763ζ7ζ32ζ7632ζ7ζ32ζ7532ζ72ζ3ζ753ζ72ζ3ζ743ζ73ζ32ζ7432ζ73ζ32ζ7432ζ73ζ32ζ7632ζ7ζ32ζ7532ζ72ζ3ζ753ζ72ζ3ζ743ζ73ζ3ζ763ζ7    complex lifted from C3×D7
ρ252-200-1+-3-1--31+-31--30000ζ7473ζ767ζ757274737677572ζ3ζ743ζ73ζ32ζ7432ζ73ζ32ζ7632ζ7ζ3ζ763ζ7ζ3ζ753ζ72ζ32ζ7532ζ7232ζ7532ζ7232ζ7432ζ7332ζ7632ζ73ζ763ζ73ζ753ζ723ζ743ζ73    complex faithful
ρ262-200-1--3-1+-31--31+-30000ζ767ζ7572ζ747376775727473ζ32ζ7632ζ7ζ3ζ763ζ7ζ3ζ753ζ72ζ32ζ7532ζ72ζ32ζ7432ζ73ζ3ζ743ζ733ζ743ζ733ζ763ζ73ζ753ζ7232ζ7532ζ7232ζ7432ζ7332ζ7632ζ7    complex faithful
ρ272-200-1+-3-1--31+-31--30000ζ767ζ7572ζ747376775727473ζ3ζ763ζ7ζ32ζ7632ζ7ζ32ζ7532ζ72ζ3ζ753ζ72ζ3ζ743ζ73ζ32ζ7432ζ7332ζ7432ζ7332ζ7632ζ732ζ7532ζ723ζ753ζ723ζ743ζ733ζ763ζ7    complex faithful
ρ282200-1--3-1+-3-1+-3-1--30000ζ767ζ7572ζ7473ζ767ζ7572ζ7473ζ32ζ7632ζ7ζ3ζ763ζ7ζ3ζ753ζ72ζ32ζ7532ζ72ζ32ζ7432ζ73ζ3ζ743ζ73ζ3ζ743ζ73ζ3ζ763ζ7ζ3ζ753ζ72ζ32ζ7532ζ72ζ32ζ7432ζ73ζ32ζ7632ζ7    complex lifted from C3×D7
ρ292200-1+-3-1--3-1--3-1+-30000ζ7572ζ7473ζ767ζ7572ζ7473ζ767ζ3ζ753ζ72ζ32ζ7532ζ72ζ32ζ7432ζ73ζ3ζ743ζ73ζ3ζ763ζ7ζ32ζ7632ζ7ζ32ζ7632ζ7ζ32ζ7532ζ72ζ32ζ7432ζ73ζ3ζ743ζ73ζ3ζ763ζ7ζ3ζ753ζ72    complex lifted from C3×D7
ρ302-200-1+-3-1--31+-31--30000ζ7572ζ7473ζ76775727473767ζ3ζ753ζ72ζ32ζ7532ζ72ζ32ζ7432ζ73ζ3ζ743ζ73ζ3ζ763ζ7ζ32ζ7632ζ732ζ7632ζ732ζ7532ζ7232ζ7432ζ733ζ743ζ733ζ763ζ73ζ753ζ72    complex faithful

Smallest permutation representation of C6×D7
On 42 points
Generators in S42
(1 34 20 27 13 41)(2 35 21 28 14 42)(3 29 15 22 8 36)(4 30 16 23 9 37)(5 31 17 24 10 38)(6 32 18 25 11 39)(7 33 19 26 12 40)
(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)(29 30 31 32 33 34 35)(36 37 38 39 40 41 42)
(1 26)(2 25)(3 24)(4 23)(5 22)(6 28)(7 27)(8 31)(9 30)(10 29)(11 35)(12 34)(13 33)(14 32)(15 38)(16 37)(17 36)(18 42)(19 41)(20 40)(21 39)

G:=sub<Sym(42)| (1,34,20,27,13,41)(2,35,21,28,14,42)(3,29,15,22,8,36)(4,30,16,23,9,37)(5,31,17,24,10,38)(6,32,18,25,11,39)(7,33,19,26,12,40), (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)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42), (1,26)(2,25)(3,24)(4,23)(5,22)(6,28)(7,27)(8,31)(9,30)(10,29)(11,35)(12,34)(13,33)(14,32)(15,38)(16,37)(17,36)(18,42)(19,41)(20,40)(21,39)>;

G:=Group( (1,34,20,27,13,41)(2,35,21,28,14,42)(3,29,15,22,8,36)(4,30,16,23,9,37)(5,31,17,24,10,38)(6,32,18,25,11,39)(7,33,19,26,12,40), (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)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42), (1,26)(2,25)(3,24)(4,23)(5,22)(6,28)(7,27)(8,31)(9,30)(10,29)(11,35)(12,34)(13,33)(14,32)(15,38)(16,37)(17,36)(18,42)(19,41)(20,40)(21,39) );

G=PermutationGroup([(1,34,20,27,13,41),(2,35,21,28,14,42),(3,29,15,22,8,36),(4,30,16,23,9,37),(5,31,17,24,10,38),(6,32,18,25,11,39),(7,33,19,26,12,40)], [(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),(29,30,31,32,33,34,35),(36,37,38,39,40,41,42)], [(1,26),(2,25),(3,24),(4,23),(5,22),(6,28),(7,27),(8,31),(9,30),(10,29),(11,35),(12,34),(13,33),(14,32),(15,38),(16,37),(17,36),(18,42),(19,41),(20,40),(21,39)])

C6×D7 is a maximal subgroup of   C21⋊D4  C3⋊D28

Matrix representation of C6×D7 in GL2(𝔽13) generated by

100
010
,
58
62
,
27
711
G:=sub<GL(2,GF(13))| [10,0,0,10],[5,6,8,2],[2,7,7,11] >;

C6×D7 in GAP, Magma, Sage, TeX

C_6\times D_7
% in TeX

G:=Group("C6xD7");
// GroupNames label

G:=SmallGroup(84,12);
// by ID

G=gap.SmallGroup(84,12);
# by ID

G:=PCGroup([4,-2,-2,-3,-7,1155]);
// Polycyclic

G:=Group<a,b,c|a^6=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C6×D7 in TeX
Character table of C6×D7 in TeX

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