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## G = C4×D7order 56 = 23·7

### Direct product of C4 and D7

Aliases: C4×D7, C282C2, D14.C2, C4Dic7, C2.1D14, Dic72C2, C14.2C22, C71(C2×C4), SmallGroup(56,4)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C4×D7
 Chief series C1 — C7 — C14 — D14 — C4×D7
 Lower central C7 — C4×D7
 Upper central C1 — C4

Generators and relations for C4×D7
G = < a,b,c | a4=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

Character table of C4×D7

 class 1 2A 2B 2C 4A 4B 4C 4D 7A 7B 7C 14A 14B 14C 28A 28B 28C 28D 28E 28F size 1 1 7 7 1 1 7 7 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 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 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 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 linear of order 2 ρ5 1 -1 -1 1 i -i -i i 1 1 1 -1 -1 -1 -i -i i i i -i linear of order 4 ρ6 1 -1 -1 1 -i i i -i 1 1 1 -1 -1 -1 i i -i -i -i i linear of order 4 ρ7 1 -1 1 -1 -i i -i i 1 1 1 -1 -1 -1 i i -i -i -i i linear of order 4 ρ8 1 -1 1 -1 i -i i -i 1 1 1 -1 -1 -1 -i -i i i i -i linear of order 4 ρ9 2 2 0 0 2 2 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 orthogonal lifted from D7 ρ10 2 2 0 0 2 2 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 orthogonal lifted from D7 ρ11 2 2 0 0 -2 -2 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ75-ζ72 orthogonal lifted from D14 ρ12 2 2 0 0 -2 -2 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ76-ζ7 orthogonal lifted from D14 ρ13 2 2 0 0 2 2 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 orthogonal lifted from D7 ρ14 2 2 0 0 -2 -2 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ74-ζ73 orthogonal lifted from D14 ρ15 2 -2 0 0 -2i 2i 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 ζ4ζ76+ζ4ζ7 ζ4ζ75+ζ4ζ72 ζ43ζ76+ζ43ζ7 ζ43ζ75+ζ43ζ72 ζ43ζ74+ζ43ζ73 ζ4ζ74+ζ4ζ73 complex faithful ρ16 2 -2 0 0 2i -2i 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 ζ43ζ75+ζ43ζ72 ζ43ζ74+ζ43ζ73 ζ4ζ75+ζ4ζ72 ζ4ζ74+ζ4ζ73 ζ4ζ76+ζ4ζ7 ζ43ζ76+ζ43ζ7 complex faithful ρ17 2 -2 0 0 2i -2i 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 ζ43ζ76+ζ43ζ7 ζ43ζ75+ζ43ζ72 ζ4ζ76+ζ4ζ7 ζ4ζ75+ζ4ζ72 ζ4ζ74+ζ4ζ73 ζ43ζ74+ζ43ζ73 complex faithful ρ18 2 -2 0 0 -2i 2i 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 ζ4ζ74+ζ4ζ73 ζ4ζ76+ζ4ζ7 ζ43ζ74+ζ43ζ73 ζ43ζ76+ζ43ζ7 ζ43ζ75+ζ43ζ72 ζ4ζ75+ζ4ζ72 complex faithful ρ19 2 -2 0 0 2i -2i 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 ζ43ζ74+ζ43ζ73 ζ43ζ76+ζ43ζ7 ζ4ζ74+ζ4ζ73 ζ4ζ76+ζ4ζ7 ζ4ζ75+ζ4ζ72 ζ43ζ75+ζ43ζ72 complex faithful ρ20 2 -2 0 0 -2i 2i 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 ζ4ζ75+ζ4ζ72 ζ4ζ74+ζ4ζ73 ζ43ζ75+ζ43ζ72 ζ43ζ74+ζ43ζ73 ζ43ζ76+ζ43ζ7 ζ4ζ76+ζ4ζ7 complex faithful

Permutation representations of C4×D7
On 28 points - transitive group 28T8
Generators in S28
(1 27 13 20)(2 28 14 21)(3 22 8 15)(4 23 9 16)(5 24 10 17)(6 25 11 18)(7 26 12 19)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)

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

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

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

G:=TransitiveGroup(28,8);

C4×D7 is a maximal subgroup of   C8⋊D7  C4○D28  D42D7  Q82D7  D21⋊C4  D70.C2  Dic72D7
C4×D7 is a maximal quotient of   C8⋊D7  Dic7⋊C4  D14⋊C4  D21⋊C4  D70.C2  Dic72D7

Matrix representation of C4×D7 in GL2(𝔽13) generated by

 8 0 0 8
,
 9 4 4 12
,
 12 0 9 1
G:=sub<GL(2,GF(13))| [8,0,0,8],[9,4,4,12],[12,9,0,1] >;

C4×D7 in GAP, Magma, Sage, TeX

C_4\times D_7
% in TeX

G:=Group("C4xD7");
// GroupNames label

G:=SmallGroup(56,4);
// by ID

G=gap.SmallGroup(56,4);
# by ID

G:=PCGroup([4,-2,-2,-2,-7,21,771]);
// Polycyclic

G:=Group<a,b,c|a^4=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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