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G = C4×D7order 56 = 23·7

Direct product of C4 and D7

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

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

C1C7 — C4×D7
C1C7C14D14 — C4×D7
C7 — C4×D7
C1C4

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

7C2
7C2
7C22
7C4
7C2×C4

Character table of C4×D7

 class 12A2B2C4A4B4C4D7A7B7C14A14B14C28A28B28C28D28E28F
 size 11771177222222222222
ρ111111111111111111111    trivial
ρ21111-1-1-1-1111111-1-1-1-1-1-1    linear of order 2
ρ311-1-1-1-111111111-1-1-1-1-1-1    linear of order 2
ρ411-1-111-1-1111111111111    linear of order 2
ρ51-1-11i-i-ii111-1-1-1-i-iiii-i    linear of order 4
ρ61-1-11-iii-i111-1-1-1ii-i-i-ii    linear of order 4
ρ71-11-1-ii-ii111-1-1-1ii-i-i-ii    linear of order 4
ρ81-11-1i-ii-i111-1-1-1-i-iiii-i    linear of order 4
ρ922002200ζ767ζ7572ζ7473ζ7572ζ767ζ7473ζ767ζ7572ζ767ζ7572ζ7473ζ7473    orthogonal lifted from D7
ρ1022002200ζ7572ζ7473ζ767ζ7473ζ7572ζ767ζ7572ζ7473ζ7572ζ7473ζ767ζ767    orthogonal lifted from D7
ρ112200-2-200ζ7473ζ767ζ7572ζ767ζ7473ζ75727473767747376775727572    orthogonal lifted from D14
ρ122200-2-200ζ7572ζ7473ζ767ζ7473ζ7572ζ7677572747375727473767767    orthogonal lifted from D14
ρ1322002200ζ7473ζ767ζ7572ζ767ζ7473ζ7572ζ7473ζ767ζ7473ζ767ζ7572ζ7572    orthogonal lifted from D7
ρ142200-2-200ζ767ζ7572ζ7473ζ7572ζ767ζ74737677572767757274737473    orthogonal lifted from D14
ρ152-200-2i2i00ζ767ζ7572ζ747375727677473ζ4ζ764ζ7ζ4ζ754ζ72ζ43ζ7643ζ7ζ43ζ7543ζ72ζ43ζ7443ζ73ζ4ζ744ζ73    complex faithful
ρ162-2002i-2i00ζ7572ζ7473ζ76774737572767ζ43ζ7543ζ72ζ43ζ7443ζ73ζ4ζ754ζ72ζ4ζ744ζ73ζ4ζ764ζ7ζ43ζ7643ζ7    complex faithful
ρ172-2002i-2i00ζ767ζ7572ζ747375727677473ζ43ζ7643ζ7ζ43ζ7543ζ72ζ4ζ764ζ7ζ4ζ754ζ72ζ4ζ744ζ73ζ43ζ7443ζ73    complex faithful
ρ182-200-2i2i00ζ7473ζ767ζ757276774737572ζ4ζ744ζ73ζ4ζ764ζ7ζ43ζ7443ζ73ζ43ζ7643ζ7ζ43ζ7543ζ72ζ4ζ754ζ72    complex faithful
ρ192-2002i-2i00ζ7473ζ767ζ757276774737572ζ43ζ7443ζ73ζ43ζ7643ζ7ζ4ζ744ζ73ζ4ζ764ζ7ζ4ζ754ζ72ζ43ζ7543ζ72    complex faithful
ρ202-200-2i2i00ζ7572ζ7473ζ76774737572767ζ4ζ754ζ72ζ4ζ744ζ73ζ43ζ7543ζ72ζ43ζ7443ζ73ζ43ζ7643ζ7ζ4ζ764ζ7    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);

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

80
08
,
94
412
,
120
91
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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