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

C7⋊C8 is a maximal subgroup of
C8×D7  C8⋊D7  C4.Dic7  D4⋊D7  D4.D7  Q8⋊D7  C7⋊Q16  C7⋊C24  C21⋊C8  C353C8  C35⋊C8  C49⋊C8  C724C8
C7⋊C8 is a maximal quotient of
C7⋊C16  C21⋊C8  C353C8  C35⋊C8  C49⋊C8  C724C8

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

Export

Subgroup lattice of C7⋊C8 in TeX
Character table of C7⋊C8 in TeX

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