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G = C5×D7  order 70 = 2·5·7

Direct product of C5 and D7

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

Aliases: C5×D7, C7⋊C10, C35⋊2C2, SmallGroup(70,2)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C7 — C5×D7
Chief series
C1 — C7 — C35 — C5×D7
Lower central
C7 — C5×D7
Upper central
C1 — C5

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

C1
7C2
C5
C7
7C10
D7
C35
C5×D7

Character table of C5×D7

 class 125A5B5C5D7A7B7C10A10B10C10D35A35B35C35D35E35F35G35H35I35J35K35L
 size 1711112227777222222222222
ρ11111111111111111111111111    trivial
ρ21-11111111-1-1-1-1111111111111    linear of order 2
ρ31-1ζ53ζ5ζ54ζ52111-ζ54-ζ5-ζ52-ζ53ζ53ζ53ζ5ζ52ζ54ζ54ζ54ζ5ζ52ζ52ζ5ζ53    linear of order 10
ρ411ζ52ζ54ζ5ζ53111ζ5ζ54ζ53ζ52ζ52ζ52ζ54ζ53ζ5ζ5ζ5ζ54ζ53ζ53ζ54ζ52    linear of order 5
ρ51-1ζ54ζ53ζ52ζ5111-ζ52-ζ53-ζ5-ζ54ζ54ζ54ζ53ζ5ζ52ζ52ζ52ζ53ζ5ζ5ζ53ζ54    linear of order 10
ρ611ζ5ζ52ζ53ζ54111ζ53ζ52ζ54ζ5ζ5ζ5ζ52ζ54ζ53ζ53ζ53ζ52ζ54ζ54ζ52ζ5    linear of order 5
ρ711ζ54ζ53ζ52ζ5111ζ52ζ53ζ5ζ54ζ54ζ54ζ53ζ5ζ52ζ52ζ52ζ53ζ5ζ5ζ53ζ54    linear of order 5
ρ811ζ53ζ5ζ54ζ52111ζ54ζ5ζ52ζ53ζ53ζ53ζ5ζ52ζ54ζ54ζ54ζ5ζ52ζ52ζ5ζ53    linear of order 5
ρ91-1ζ5ζ52ζ53ζ54111-ζ53-ζ52-ζ54-ζ5ζ5ζ5ζ52ζ54ζ53ζ53ζ53ζ52ζ54ζ54ζ52ζ5    linear of order 10
ρ101-1ζ52ζ54ζ5ζ53111-ζ5-ζ54-ζ53-ζ52ζ52ζ52ζ54ζ53ζ5ζ5ζ5ζ54ζ53ζ53ζ54ζ52    linear of order 10
ρ11202222ζ76+ζ7ζ75+ζ72ζ74+ζ730000ζ74+ζ73ζ75+ζ72ζ75+ζ72ζ76+ζ7ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ74+ζ73ζ76+ζ7    orthogonal lifted from D7
ρ12202222ζ75+ζ72ζ74+ζ73ζ76+ζ70000ζ76+ζ7ζ74+ζ73ζ74+ζ73ζ75+ζ72ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ76+ζ7ζ75+ζ72    orthogonal lifted from D7
ρ13202222ζ74+ζ73ζ76+ζ7ζ75+ζ720000ζ75+ζ72ζ76+ζ7ζ76+ζ7ζ74+ζ73ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ74+ζ73ζ76+ζ7ζ75+ζ72ζ75+ζ72ζ74+ζ73    orthogonal lifted from D7
ρ14202ζ532ζ52ζ542ζ52ζ74+ζ73ζ76+ζ7ζ75+ζ720000ζ53ζ75+ζ53ζ72ζ53ζ76+ζ53ζ7ζ5ζ76+ζ5ζ7ζ52ζ74+ζ52ζ73ζ54ζ74+ζ54ζ73ζ54ζ76+ζ54ζ7ζ54ζ75+ζ54ζ72ζ5ζ74+ζ5ζ73ζ52ζ76+ζ52ζ7ζ52ζ75+ζ52ζ72ζ5ζ75+ζ5ζ72ζ53ζ74+ζ53ζ73    complex faithful
ρ15202ζ532ζ52ζ542ζ52ζ76+ζ7ζ75+ζ72ζ74+ζ730000ζ53ζ74+ζ53ζ73ζ53ζ75+ζ53ζ72ζ5ζ75+ζ5ζ72ζ52ζ76+ζ52ζ7ζ54ζ76+ζ54ζ7ζ54ζ75+ζ54ζ72ζ54ζ74+ζ54ζ73ζ5ζ76+ζ5ζ7ζ52ζ75+ζ52ζ72ζ52ζ74+ζ52ζ73ζ5ζ74+ζ5ζ73ζ53ζ76+ζ53ζ7    complex faithful
ρ16202ζ542ζ532ζ522ζ5ζ75+ζ72ζ74+ζ73ζ76+ζ70000ζ54ζ76+ζ54ζ7ζ54ζ74+ζ54ζ73ζ53ζ74+ζ53ζ73ζ5ζ75+ζ5ζ72ζ52ζ75+ζ52ζ72ζ52ζ74+ζ52ζ73ζ52ζ76+ζ52ζ7ζ53ζ75+ζ53ζ72ζ5ζ74+ζ5ζ73ζ5ζ76+ζ5ζ7ζ53ζ76+ζ53ζ7ζ54ζ75+ζ54ζ72    complex faithful
ρ17202ζ52ζ522ζ532ζ54ζ74+ζ73ζ76+ζ7ζ75+ζ720000ζ5ζ75+ζ5ζ72ζ5ζ76+ζ5ζ7ζ52ζ76+ζ52ζ7ζ54ζ74+ζ54ζ73ζ53ζ74+ζ53ζ73ζ53ζ76+ζ53ζ7ζ53ζ75+ζ53ζ72ζ52ζ74+ζ52ζ73ζ54ζ76+ζ54ζ7ζ54ζ75+ζ54ζ72ζ52ζ75+ζ52ζ72ζ5ζ74+ζ5ζ73    complex faithful
ρ18202ζ52ζ522ζ532ζ54ζ76+ζ7ζ75+ζ72ζ74+ζ730000ζ5ζ74+ζ5ζ73ζ5ζ75+ζ5ζ72ζ52ζ75+ζ52ζ72ζ54ζ76+ζ54ζ7ζ53ζ76+ζ53ζ7ζ53ζ75+ζ53ζ72ζ53ζ74+ζ53ζ73ζ52ζ76+ζ52ζ7ζ54ζ75+ζ54ζ72ζ54ζ74+ζ54ζ73ζ52ζ74+ζ52ζ73ζ5ζ76+ζ5ζ7    complex faithful
ρ19202ζ532ζ52ζ542ζ52ζ75+ζ72ζ74+ζ73ζ76+ζ70000ζ53ζ76+ζ53ζ7ζ53ζ74+ζ53ζ73ζ5ζ74+ζ5ζ73ζ52ζ75+ζ52ζ72ζ54ζ75+ζ54ζ72ζ54ζ74+ζ54ζ73ζ54ζ76+ζ54ζ7ζ5ζ75+ζ5ζ72ζ52ζ74+ζ52ζ73ζ52ζ76+ζ52ζ7ζ5ζ76+ζ5ζ7ζ53ζ75+ζ53ζ72    complex faithful
ρ20202ζ52ζ522ζ532ζ54ζ75+ζ72ζ74+ζ73ζ76+ζ70000ζ5ζ76+ζ5ζ7ζ5ζ74+ζ5ζ73ζ52ζ74+ζ52ζ73ζ54ζ75+ζ54ζ72ζ53ζ75+ζ53ζ72ζ53ζ74+ζ53ζ73ζ53ζ76+ζ53ζ7ζ52ζ75+ζ52ζ72ζ54ζ74+ζ54ζ73ζ54ζ76+ζ54ζ7ζ52ζ76+ζ52ζ7ζ5ζ75+ζ5ζ72    complex faithful
ρ21202ζ542ζ532ζ522ζ5ζ74+ζ73ζ76+ζ7ζ75+ζ720000ζ54ζ75+ζ54ζ72ζ54ζ76+ζ54ζ7ζ53ζ76+ζ53ζ7ζ5ζ74+ζ5ζ73ζ52ζ74+ζ52ζ73ζ52ζ76+ζ52ζ7ζ52ζ75+ζ52ζ72ζ53ζ74+ζ53ζ73ζ5ζ76+ζ5ζ7ζ5ζ75+ζ5ζ72ζ53ζ75+ζ53ζ72ζ54ζ74+ζ54ζ73    complex faithful
ρ22202ζ542ζ532ζ522ζ5ζ76+ζ7ζ75+ζ72ζ74+ζ730000ζ54ζ74+ζ54ζ73ζ54ζ75+ζ54ζ72ζ53ζ75+ζ53ζ72ζ5ζ76+ζ5ζ7ζ52ζ76+ζ52ζ7ζ52ζ75+ζ52ζ72ζ52ζ74+ζ52ζ73ζ53ζ76+ζ53ζ7ζ5ζ75+ζ5ζ72ζ5ζ74+ζ5ζ73ζ53ζ74+ζ53ζ73ζ54ζ76+ζ54ζ7    complex faithful
ρ23202ζ522ζ542ζ52ζ53ζ74+ζ73ζ76+ζ7ζ75+ζ720000ζ52ζ75+ζ52ζ72ζ52ζ76+ζ52ζ7ζ54ζ76+ζ54ζ7ζ53ζ74+ζ53ζ73ζ5ζ74+ζ5ζ73ζ5ζ76+ζ5ζ7ζ5ζ75+ζ5ζ72ζ54ζ74+ζ54ζ73ζ53ζ76+ζ53ζ7ζ53ζ75+ζ53ζ72ζ54ζ75+ζ54ζ72ζ52ζ74+ζ52ζ73    complex faithful
ρ24202ζ522ζ542ζ52ζ53ζ75+ζ72ζ74+ζ73ζ76+ζ70000ζ52ζ76+ζ52ζ7ζ52ζ74+ζ52ζ73ζ54ζ74+ζ54ζ73ζ53ζ75+ζ53ζ72ζ5ζ75+ζ5ζ72ζ5ζ74+ζ5ζ73ζ5ζ76+ζ5ζ7ζ54ζ75+ζ54ζ72ζ53ζ74+ζ53ζ73ζ53ζ76+ζ53ζ7ζ54ζ76+ζ54ζ7ζ52ζ75+ζ52ζ72    complex faithful
ρ25202ζ522ζ542ζ52ζ53ζ76+ζ7ζ75+ζ72ζ74+ζ730000ζ52ζ74+ζ52ζ73ζ52ζ75+ζ52ζ72ζ54ζ75+ζ54ζ72ζ53ζ76+ζ53ζ7ζ5ζ76+ζ5ζ7ζ5ζ75+ζ5ζ72ζ5ζ74+ζ5ζ73ζ54ζ76+ζ54ζ7ζ53ζ75+ζ53ζ72ζ53ζ74+ζ53ζ73ζ54ζ74+ζ54ζ73ζ52ζ76+ζ52ζ7    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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