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G = C5×D7order 70 = 2·5·7

Direct product of C5 and D7

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

Aliases: C5×D7, C7⋊C10, C352C2, SmallGroup(70,2)

Series: Derived Chief Lower central Upper central

C1C7 — C5×D7
C1C7C35 — C5×D7
C7 — C5×D7
C1C5

Generators and relations for C5×D7
 G = < a,b,c | a5=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

7C2
7C10

Character table of C5×D7

 class 125A5B5C5D7A7B7C10A10B10C10D35A35B35C35D35E35F35G35H35I35J35K35L
 size 1711112227777222222222222
ρ11111111111111111111111111    trivial
ρ21-11111111-1-1-1-1111111111111    linear of order 2
ρ31-1ζ53ζ5ζ54ζ521115455253ζ53ζ53ζ5ζ52ζ54ζ54ζ54ζ5ζ52ζ52ζ5ζ53    linear of order 10
ρ411ζ52ζ54ζ5ζ53111ζ5ζ54ζ53ζ52ζ52ζ52ζ54ζ53ζ5ζ5ζ5ζ54ζ53ζ53ζ54ζ52    linear of order 5
ρ51-1ζ54ζ53ζ52ζ51115253554ζ54ζ54ζ53ζ5ζ52ζ52ζ52ζ53ζ5ζ5ζ53ζ54    linear of order 10
ρ611ζ5ζ52ζ53ζ54111ζ53ζ52ζ54ζ5ζ5ζ5ζ52ζ54ζ53ζ53ζ53ζ52ζ54ζ54ζ52ζ5    linear of order 5
ρ711ζ54ζ53ζ52ζ5111ζ52ζ53ζ5ζ54ζ54ζ54ζ53ζ5ζ52ζ52ζ52ζ53ζ5ζ5ζ53ζ54    linear of order 5
ρ811ζ53ζ5ζ54ζ52111ζ54ζ5ζ52ζ53ζ53ζ53ζ5ζ52ζ54ζ54ζ54ζ5ζ52ζ52ζ5ζ53    linear of order 5
ρ91-1ζ5ζ52ζ53ζ541115352545ζ5ζ5ζ52ζ54ζ53ζ53ζ53ζ52ζ54ζ54ζ52ζ5    linear of order 10
ρ101-1ζ52ζ54ζ5ζ531115545352ζ52ζ52ζ54ζ53ζ5ζ5ζ5ζ54ζ53ζ53ζ54ζ52    linear of order 10
ρ11202222ζ767ζ7572ζ74730000ζ7473ζ7572ζ7572ζ767ζ767ζ7572ζ7473ζ767ζ7572ζ7473ζ7473ζ767    orthogonal lifted from D7
ρ12202222ζ7572ζ7473ζ7670000ζ767ζ7473ζ7473ζ7572ζ7572ζ7473ζ767ζ7572ζ7473ζ767ζ767ζ7572    orthogonal lifted from D7
ρ13202222ζ7473ζ767ζ75720000ζ7572ζ767ζ767ζ7473ζ7473ζ767ζ7572ζ7473ζ767ζ7572ζ7572ζ7473    orthogonal lifted from D7
ρ14205355452ζ7473ζ767ζ75720000ζ53ζ7553ζ72ζ53ζ7653ζ7ζ5ζ765ζ7ζ52ζ7452ζ73ζ54ζ7454ζ73ζ54ζ7654ζ7ζ54ζ7554ζ72ζ5ζ745ζ73ζ52ζ7652ζ7ζ52ζ7552ζ72ζ5ζ755ζ72ζ53ζ7453ζ73    complex faithful
ρ15205355452ζ767ζ7572ζ74730000ζ53ζ7453ζ73ζ53ζ7553ζ72ζ5ζ755ζ72ζ52ζ7652ζ7ζ54ζ7654ζ7ζ54ζ7554ζ72ζ54ζ7454ζ73ζ5ζ765ζ7ζ52ζ7552ζ72ζ52ζ7452ζ73ζ5ζ745ζ73ζ53ζ7653ζ7    complex faithful
ρ16205453525ζ7572ζ7473ζ7670000ζ54ζ7654ζ7ζ54ζ7454ζ73ζ53ζ7453ζ73ζ5ζ755ζ72ζ52ζ7552ζ72ζ52ζ7452ζ73ζ52ζ7652ζ7ζ53ζ7553ζ72ζ5ζ745ζ73ζ5ζ765ζ7ζ53ζ7653ζ7ζ54ζ7554ζ72    complex faithful
ρ17205525354ζ7473ζ767ζ75720000ζ5ζ755ζ72ζ5ζ765ζ7ζ52ζ7652ζ7ζ54ζ7454ζ73ζ53ζ7453ζ73ζ53ζ7653ζ7ζ53ζ7553ζ72ζ52ζ7452ζ73ζ54ζ7654ζ7ζ54ζ7554ζ72ζ52ζ7552ζ72ζ5ζ745ζ73    complex faithful
ρ18205525354ζ767ζ7572ζ74730000ζ5ζ745ζ73ζ5ζ755ζ72ζ52ζ7552ζ72ζ54ζ7654ζ7ζ53ζ7653ζ7ζ53ζ7553ζ72ζ53ζ7453ζ73ζ52ζ7652ζ7ζ54ζ7554ζ72ζ54ζ7454ζ73ζ52ζ7452ζ73ζ5ζ765ζ7    complex faithful
ρ19205355452ζ7572ζ7473ζ7670000ζ53ζ7653ζ7ζ53ζ7453ζ73ζ5ζ745ζ73ζ52ζ7552ζ72ζ54ζ7554ζ72ζ54ζ7454ζ73ζ54ζ7654ζ7ζ5ζ755ζ72ζ52ζ7452ζ73ζ52ζ7652ζ7ζ5ζ765ζ7ζ53ζ7553ζ72    complex faithful
ρ20205525354ζ7572ζ7473ζ7670000ζ5ζ765ζ7ζ5ζ745ζ73ζ52ζ7452ζ73ζ54ζ7554ζ72ζ53ζ7553ζ72ζ53ζ7453ζ73ζ53ζ7653ζ7ζ52ζ7552ζ72ζ54ζ7454ζ73ζ54ζ7654ζ7ζ52ζ7652ζ7ζ5ζ755ζ72    complex faithful
ρ21205453525ζ7473ζ767ζ75720000ζ54ζ7554ζ72ζ54ζ7654ζ7ζ53ζ7653ζ7ζ5ζ745ζ73ζ52ζ7452ζ73ζ52ζ7652ζ7ζ52ζ7552ζ72ζ53ζ7453ζ73ζ5ζ765ζ7ζ5ζ755ζ72ζ53ζ7553ζ72ζ54ζ7454ζ73    complex faithful
ρ22205453525ζ767ζ7572ζ74730000ζ54ζ7454ζ73ζ54ζ7554ζ72ζ53ζ7553ζ72ζ5ζ765ζ7ζ52ζ7652ζ7ζ52ζ7552ζ72ζ52ζ7452ζ73ζ53ζ7653ζ7ζ5ζ755ζ72ζ5ζ745ζ73ζ53ζ7453ζ73ζ54ζ7654ζ7    complex faithful
ρ23205254553ζ7473ζ767ζ75720000ζ52ζ7552ζ72ζ52ζ7652ζ7ζ54ζ7654ζ7ζ53ζ7453ζ73ζ5ζ745ζ73ζ5ζ765ζ7ζ5ζ755ζ72ζ54ζ7454ζ73ζ53ζ7653ζ7ζ53ζ7553ζ72ζ54ζ7554ζ72ζ52ζ7452ζ73    complex faithful
ρ24205254553ζ7572ζ7473ζ7670000ζ52ζ7652ζ7ζ52ζ7452ζ73ζ54ζ7454ζ73ζ53ζ7553ζ72ζ5ζ755ζ72ζ5ζ745ζ73ζ5ζ765ζ7ζ54ζ7554ζ72ζ53ζ7453ζ73ζ53ζ7653ζ7ζ54ζ7654ζ7ζ52ζ7552ζ72    complex faithful
ρ25205254553ζ767ζ7572ζ74730000ζ52ζ7452ζ73ζ52ζ7552ζ72ζ54ζ7554ζ72ζ53ζ7653ζ7ζ5ζ765ζ7ζ5ζ755ζ72ζ5ζ745ζ73ζ54ζ7654ζ7ζ53ζ7553ζ72ζ53ζ7453ζ73ζ54ζ7454ζ73ζ52ζ7652ζ7    complex faithful

Smallest permutation representation of C5×D7
On 35 points
Generators in S35
(1 34 27 20 13)(2 35 28 21 14)(3 29 22 15 8)(4 30 23 16 9)(5 31 24 17 10)(6 32 25 18 11)(7 33 26 19 12)
(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)
(1 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 31)(32 35)(33 34)

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

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

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

Matrix representation of C5×D7 in GL2(𝔽41) generated by

160
016
,
151
2540
,
400
161
G:=sub<GL(2,GF(41))| [16,0,0,16],[15,25,1,40],[40,16,0,1] >;

C5×D7 in GAP, Magma, Sage, TeX

C_5\times D_7
% in TeX

G:=Group("C5xD7");
// GroupNames label

G:=SmallGroup(70,2);
// by ID

G=gap.SmallGroup(70,2);
# by ID

G:=PCGroup([3,-2,-5,-7,542]);
// Polycyclic

G:=Group<a,b,c|a^5=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×D7 in TeX
Character table of C5×D7 in TeX

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