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G = C7⋊C8order 56 = 23·7

The semidirect product of C7 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C7⋊C8, C14.C4, C4.2D7, C2.Dic7, C28.2C2, SmallGroup(56,1)

Series: Derived Chief Lower central Upper central

C1C7 — C7⋊C8
C1C7C14C28 — C7⋊C8
C7 — C7⋊C8
C1C4

Generators and relations for C7⋊C8
 G = < a,b | a7=b8=1, bab-1=a-1 >

7C8

Character table of C7⋊C8

 class 124A4B7A7B7C8A8B8C8D14A14B14C28A28B28C28D28E28F
 size 11112227777222222222
ρ111111111111111111111    trivial
ρ21111111-1-1-1-1111111111    linear of order 2
ρ311-1-1111i-ii-i111-1-1-1-1-1-1    linear of order 4
ρ411-1-1111-ii-ii111-1-1-1-1-1-1    linear of order 4
ρ51-1-ii111ζ83ζ8ζ87ζ85-1-1-1i-i-i-iii    linear of order 8
ρ61-1i-i111ζ8ζ83ζ85ζ87-1-1-1-iiii-i-i    linear of order 8
ρ71-1-ii111ζ87ζ85ζ83ζ8-1-1-1i-i-i-iii    linear of order 8
ρ81-1i-i111ζ85ζ87ζ8ζ83-1-1-1-iiii-i-i    linear of order 8
ρ92222ζ767ζ7572ζ74730000ζ767ζ7572ζ7473ζ7473ζ7473ζ767ζ7572ζ767ζ7572    orthogonal lifted from D7
ρ102222ζ7473ζ767ζ75720000ζ7473ζ767ζ7572ζ7572ζ7572ζ7473ζ767ζ7473ζ767    orthogonal lifted from D7
ρ112222ζ7572ζ7473ζ7670000ζ7572ζ7473ζ767ζ767ζ767ζ7572ζ7473ζ7572ζ7473    orthogonal lifted from D7
ρ1222-2-2ζ7572ζ7473ζ7670000ζ7572ζ7473ζ7677677677572747375727473    symplectic lifted from Dic7, Schur index 2
ρ1322-2-2ζ767ζ7572ζ74730000ζ767ζ7572ζ74737473747376775727677572    symplectic lifted from Dic7, Schur index 2
ρ1422-2-2ζ7473ζ767ζ75720000ζ7473ζ767ζ75727572757274737677473767    symplectic lifted from Dic7, Schur index 2
ρ152-22i-2iζ767ζ7572ζ7473000076775727473ζ43ζ7443ζ73ζ4ζ744ζ73ζ4ζ764ζ7ζ4ζ754ζ72ζ43ζ7643ζ7ζ43ζ7543ζ72    complex faithful
ρ162-22i-2iζ7572ζ7473ζ767000075727473767ζ43ζ7643ζ7ζ4ζ764ζ7ζ4ζ754ζ72ζ4ζ744ζ73ζ43ζ7543ζ72ζ43ζ7443ζ73    complex faithful
ρ172-2-2i2iζ767ζ7572ζ7473000076775727473ζ4ζ744ζ73ζ43ζ7443ζ73ζ43ζ7643ζ7ζ43ζ7543ζ72ζ4ζ764ζ7ζ4ζ754ζ72    complex faithful
ρ182-2-2i2iζ7473ζ767ζ7572000074737677572ζ4ζ754ζ72ζ43ζ7543ζ72ζ43ζ7443ζ73ζ43ζ7643ζ7ζ4ζ744ζ73ζ4ζ764ζ7    complex faithful
ρ192-22i-2iζ7473ζ767ζ7572000074737677572ζ43ζ7543ζ72ζ4ζ754ζ72ζ4ζ744ζ73ζ4ζ764ζ7ζ43ζ7443ζ73ζ43ζ7643ζ7    complex faithful
ρ202-2-2i2iζ7572ζ7473ζ767000075727473767ζ4ζ764ζ7ζ43ζ7643ζ7ζ43ζ7543ζ72ζ43ζ7443ζ73ζ4ζ754ζ72ζ4ζ744ζ73    complex faithful

Smallest permutation representation of C7⋊C8
Regular action on 56 points
Generators in S56
(1 49 17 48 37 13 31)(2 32 14 38 41 18 50)(3 51 19 42 39 15 25)(4 26 16 40 43 20 52)(5 53 21 44 33 9 27)(6 28 10 34 45 22 54)(7 55 23 46 35 11 29)(8 30 12 36 47 24 56)
(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 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)

G:=sub<Sym(56)| (1,49,17,48,37,13,31)(2,32,14,38,41,18,50)(3,51,19,42,39,15,25)(4,26,16,40,43,20,52)(5,53,21,44,33,9,27)(6,28,10,34,45,22,54)(7,55,23,46,35,11,29)(8,30,12,36,47,24,56), (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)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)>;

G:=Group( (1,49,17,48,37,13,31)(2,32,14,38,41,18,50)(3,51,19,42,39,15,25)(4,26,16,40,43,20,52)(5,53,21,44,33,9,27)(6,28,10,34,45,22,54)(7,55,23,46,35,11,29)(8,30,12,36,47,24,56), (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)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56) );

G=PermutationGroup([(1,49,17,48,37,13,31),(2,32,14,38,41,18,50),(3,51,19,42,39,15,25),(4,26,16,40,43,20,52),(5,53,21,44,33,9,27),(6,28,10,34,45,22,54),(7,55,23,46,35,11,29),(8,30,12,36,47,24,56)], [(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),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56)])

Matrix representation of C7⋊C8 in GL2(𝔽13) generated by

54
65
,
08
10
G:=sub<GL(2,GF(13))| [5,6,4,5],[0,1,8,0] >;

C7⋊C8 in GAP, Magma, Sage, TeX

C_7\rtimes C_8
% in TeX

G:=Group("C7:C8");
// GroupNames label

G:=SmallGroup(56,1);
// by ID

G=gap.SmallGroup(56,1);
# by ID

G:=PCGroup([4,-2,-2,-2,-7,8,21,771]);
// Polycyclic

G:=Group<a,b|a^7=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

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