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## G = C7×D5order 70 = 2·5·7

### Direct product of C7 and D5

Aliases: C7×D5, C5⋊C14, C353C2, SmallGroup(70,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C7×D5
 Chief series C1 — C5 — C35 — C7×D5
 Lower central C5 — C7×D5
 Upper central C1 — C7

Generators and relations for C7×D5
G = < a,b,c | a7=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

Character table of C7×D5

 class 1 2 5A 5B 7A 7B 7C 7D 7E 7F 14A 14B 14C 14D 14E 14F 35A 35B 35C 35D 35E 35F 35G 35H 35I 35J 35K 35L size 1 5 2 2 1 1 1 1 1 1 5 5 5 5 5 5 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ3 1 1 1 1 ζ72 ζ73 ζ74 ζ75 ζ76 ζ7 ζ76 ζ72 ζ73 ζ74 ζ75 ζ7 ζ75 ζ75 ζ72 ζ76 ζ76 ζ7 ζ73 ζ73 ζ7 ζ74 ζ74 ζ72 linear of order 7 ρ4 1 1 1 1 ζ75 ζ74 ζ73 ζ72 ζ7 ζ76 ζ7 ζ75 ζ74 ζ73 ζ72 ζ76 ζ72 ζ72 ζ75 ζ7 ζ7 ζ76 ζ74 ζ74 ζ76 ζ73 ζ73 ζ75 linear of order 7 ρ5 1 -1 1 1 ζ7 ζ75 ζ72 ζ76 ζ73 ζ74 -ζ73 -ζ7 -ζ75 -ζ72 -ζ76 -ζ74 ζ76 ζ76 ζ7 ζ73 ζ73 ζ74 ζ75 ζ75 ζ74 ζ72 ζ72 ζ7 linear of order 14 ρ6 1 -1 1 1 ζ72 ζ73 ζ74 ζ75 ζ76 ζ7 -ζ76 -ζ72 -ζ73 -ζ74 -ζ75 -ζ7 ζ75 ζ75 ζ72 ζ76 ζ76 ζ7 ζ73 ζ73 ζ7 ζ74 ζ74 ζ72 linear of order 14 ρ7 1 -1 1 1 ζ73 ζ7 ζ76 ζ74 ζ72 ζ75 -ζ72 -ζ73 -ζ7 -ζ76 -ζ74 -ζ75 ζ74 ζ74 ζ73 ζ72 ζ72 ζ75 ζ7 ζ7 ζ75 ζ76 ζ76 ζ73 linear of order 14 ρ8 1 1 1 1 ζ7 ζ75 ζ72 ζ76 ζ73 ζ74 ζ73 ζ7 ζ75 ζ72 ζ76 ζ74 ζ76 ζ76 ζ7 ζ73 ζ73 ζ74 ζ75 ζ75 ζ74 ζ72 ζ72 ζ7 linear of order 7 ρ9 1 -1 1 1 ζ74 ζ76 ζ7 ζ73 ζ75 ζ72 -ζ75 -ζ74 -ζ76 -ζ7 -ζ73 -ζ72 ζ73 ζ73 ζ74 ζ75 ζ75 ζ72 ζ76 ζ76 ζ72 ζ7 ζ7 ζ74 linear of order 14 ρ10 1 1 1 1 ζ76 ζ72 ζ75 ζ7 ζ74 ζ73 ζ74 ζ76 ζ72 ζ75 ζ7 ζ73 ζ7 ζ7 ζ76 ζ74 ζ74 ζ73 ζ72 ζ72 ζ73 ζ75 ζ75 ζ76 linear of order 7 ρ11 1 -1 1 1 ζ76 ζ72 ζ75 ζ7 ζ74 ζ73 -ζ74 -ζ76 -ζ72 -ζ75 -ζ7 -ζ73 ζ7 ζ7 ζ76 ζ74 ζ74 ζ73 ζ72 ζ72 ζ73 ζ75 ζ75 ζ76 linear of order 14 ρ12 1 -1 1 1 ζ75 ζ74 ζ73 ζ72 ζ7 ζ76 -ζ7 -ζ75 -ζ74 -ζ73 -ζ72 -ζ76 ζ72 ζ72 ζ75 ζ7 ζ7 ζ76 ζ74 ζ74 ζ76 ζ73 ζ73 ζ75 linear of order 14 ρ13 1 1 1 1 ζ73 ζ7 ζ76 ζ74 ζ72 ζ75 ζ72 ζ73 ζ7 ζ76 ζ74 ζ75 ζ74 ζ74 ζ73 ζ72 ζ72 ζ75 ζ7 ζ7 ζ75 ζ76 ζ76 ζ73 linear of order 7 ρ14 1 1 1 1 ζ74 ζ76 ζ7 ζ73 ζ75 ζ72 ζ75 ζ74 ζ76 ζ7 ζ73 ζ72 ζ73 ζ73 ζ74 ζ75 ζ75 ζ72 ζ76 ζ76 ζ72 ζ7 ζ7 ζ74 linear of order 7 ρ15 2 0 -1-√5/2 -1+√5/2 2 2 2 2 2 2 0 0 0 0 0 0 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ16 2 0 -1+√5/2 -1-√5/2 2 2 2 2 2 2 0 0 0 0 0 0 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ17 2 0 -1+√5/2 -1-√5/2 2ζ76 2ζ72 2ζ75 2ζ7 2ζ74 2ζ73 0 0 0 0 0 0 ζ53ζ7+ζ52ζ7 ζ54ζ7+ζ5ζ7 ζ54ζ76+ζ5ζ76 ζ53ζ74+ζ52ζ74 ζ54ζ74+ζ5ζ74 ζ54ζ73+ζ5ζ73 ζ53ζ72+ζ52ζ72 ζ54ζ72+ζ5ζ72 ζ53ζ73+ζ52ζ73 ζ53ζ75+ζ52ζ75 ζ54ζ75+ζ5ζ75 ζ53ζ76+ζ52ζ76 complex faithful ρ18 2 0 -1-√5/2 -1+√5/2 2ζ72 2ζ73 2ζ74 2ζ75 2ζ76 2ζ7 0 0 0 0 0 0 ζ54ζ75+ζ5ζ75 ζ53ζ75+ζ52ζ75 ζ53ζ72+ζ52ζ72 ζ54ζ76+ζ5ζ76 ζ53ζ76+ζ52ζ76 ζ53ζ7+ζ52ζ7 ζ54ζ73+ζ5ζ73 ζ53ζ73+ζ52ζ73 ζ54ζ7+ζ5ζ7 ζ54ζ74+ζ5ζ74 ζ53ζ74+ζ52ζ74 ζ54ζ72+ζ5ζ72 complex faithful ρ19 2 0 -1-√5/2 -1+√5/2 2ζ75 2ζ74 2ζ73 2ζ72 2ζ7 2ζ76 0 0 0 0 0 0 ζ54ζ72+ζ5ζ72 ζ53ζ72+ζ52ζ72 ζ53ζ75+ζ52ζ75 ζ54ζ7+ζ5ζ7 ζ53ζ7+ζ52ζ7 ζ53ζ76+ζ52ζ76 ζ54ζ74+ζ5ζ74 ζ53ζ74+ζ52ζ74 ζ54ζ76+ζ5ζ76 ζ54ζ73+ζ5ζ73 ζ53ζ73+ζ52ζ73 ζ54ζ75+ζ5ζ75 complex faithful ρ20 2 0 -1+√5/2 -1-√5/2 2ζ74 2ζ76 2ζ7 2ζ73 2ζ75 2ζ72 0 0 0 0 0 0 ζ53ζ73+ζ52ζ73 ζ54ζ73+ζ5ζ73 ζ54ζ74+ζ5ζ74 ζ53ζ75+ζ52ζ75 ζ54ζ75+ζ5ζ75 ζ54ζ72+ζ5ζ72 ζ53ζ76+ζ52ζ76 ζ54ζ76+ζ5ζ76 ζ53ζ72+ζ52ζ72 ζ53ζ7+ζ52ζ7 ζ54ζ7+ζ5ζ7 ζ53ζ74+ζ52ζ74 complex faithful ρ21 2 0 -1+√5/2 -1-√5/2 2ζ73 2ζ7 2ζ76 2ζ74 2ζ72 2ζ75 0 0 0 0 0 0 ζ53ζ74+ζ52ζ74 ζ54ζ74+ζ5ζ74 ζ54ζ73+ζ5ζ73 ζ53ζ72+ζ52ζ72 ζ54ζ72+ζ5ζ72 ζ54ζ75+ζ5ζ75 ζ53ζ7+ζ52ζ7 ζ54ζ7+ζ5ζ7 ζ53ζ75+ζ52ζ75 ζ53ζ76+ζ52ζ76 ζ54ζ76+ζ5ζ76 ζ53ζ73+ζ52ζ73 complex faithful ρ22 2 0 -1-√5/2 -1+√5/2 2ζ7 2ζ75 2ζ72 2ζ76 2ζ73 2ζ74 0 0 0 0 0 0 ζ54ζ76+ζ5ζ76 ζ53ζ76+ζ52ζ76 ζ53ζ7+ζ52ζ7 ζ54ζ73+ζ5ζ73 ζ53ζ73+ζ52ζ73 ζ53ζ74+ζ52ζ74 ζ54ζ75+ζ5ζ75 ζ53ζ75+ζ52ζ75 ζ54ζ74+ζ5ζ74 ζ54ζ72+ζ5ζ72 ζ53ζ72+ζ52ζ72 ζ54ζ7+ζ5ζ7 complex faithful ρ23 2 0 -1-√5/2 -1+√5/2 2ζ74 2ζ76 2ζ7 2ζ73 2ζ75 2ζ72 0 0 0 0 0 0 ζ54ζ73+ζ5ζ73 ζ53ζ73+ζ52ζ73 ζ53ζ74+ζ52ζ74 ζ54ζ75+ζ5ζ75 ζ53ζ75+ζ52ζ75 ζ53ζ72+ζ52ζ72 ζ54ζ76+ζ5ζ76 ζ53ζ76+ζ52ζ76 ζ54ζ72+ζ5ζ72 ζ54ζ7+ζ5ζ7 ζ53ζ7+ζ52ζ7 ζ54ζ74+ζ5ζ74 complex faithful ρ24 2 0 -1+√5/2 -1-√5/2 2ζ7 2ζ75 2ζ72 2ζ76 2ζ73 2ζ74 0 0 0 0 0 0 ζ53ζ76+ζ52ζ76 ζ54ζ76+ζ5ζ76 ζ54ζ7+ζ5ζ7 ζ53ζ73+ζ52ζ73 ζ54ζ73+ζ5ζ73 ζ54ζ74+ζ5ζ74 ζ53ζ75+ζ52ζ75 ζ54ζ75+ζ5ζ75 ζ53ζ74+ζ52ζ74 ζ53ζ72+ζ52ζ72 ζ54ζ72+ζ5ζ72 ζ53ζ7+ζ52ζ7 complex faithful ρ25 2 0 -1-√5/2 -1+√5/2 2ζ76 2ζ72 2ζ75 2ζ7 2ζ74 2ζ73 0 0 0 0 0 0 ζ54ζ7+ζ5ζ7 ζ53ζ7+ζ52ζ7 ζ53ζ76+ζ52ζ76 ζ54ζ74+ζ5ζ74 ζ53ζ74+ζ52ζ74 ζ53ζ73+ζ52ζ73 ζ54ζ72+ζ5ζ72 ζ53ζ72+ζ52ζ72 ζ54ζ73+ζ5ζ73 ζ54ζ75+ζ5ζ75 ζ53ζ75+ζ52ζ75 ζ54ζ76+ζ5ζ76 complex faithful ρ26 2 0 -1+√5/2 -1-√5/2 2ζ72 2ζ73 2ζ74 2ζ75 2ζ76 2ζ7 0 0 0 0 0 0 ζ53ζ75+ζ52ζ75 ζ54ζ75+ζ5ζ75 ζ54ζ72+ζ5ζ72 ζ53ζ76+ζ52ζ76 ζ54ζ76+ζ5ζ76 ζ54ζ7+ζ5ζ7 ζ53ζ73+ζ52ζ73 ζ54ζ73+ζ5ζ73 ζ53ζ7+ζ52ζ7 ζ53ζ74+ζ52ζ74 ζ54ζ74+ζ5ζ74 ζ53ζ72+ζ52ζ72 complex faithful ρ27 2 0 -1-√5/2 -1+√5/2 2ζ73 2ζ7 2ζ76 2ζ74 2ζ72 2ζ75 0 0 0 0 0 0 ζ54ζ74+ζ5ζ74 ζ53ζ74+ζ52ζ74 ζ53ζ73+ζ52ζ73 ζ54ζ72+ζ5ζ72 ζ53ζ72+ζ52ζ72 ζ53ζ75+ζ52ζ75 ζ54ζ7+ζ5ζ7 ζ53ζ7+ζ52ζ7 ζ54ζ75+ζ5ζ75 ζ54ζ76+ζ5ζ76 ζ53ζ76+ζ52ζ76 ζ54ζ73+ζ5ζ73 complex faithful ρ28 2 0 -1+√5/2 -1-√5/2 2ζ75 2ζ74 2ζ73 2ζ72 2ζ7 2ζ76 0 0 0 0 0 0 ζ53ζ72+ζ52ζ72 ζ54ζ72+ζ5ζ72 ζ54ζ75+ζ5ζ75 ζ53ζ7+ζ52ζ7 ζ54ζ7+ζ5ζ7 ζ54ζ76+ζ5ζ76 ζ53ζ74+ζ52ζ74 ζ54ζ74+ζ5ζ74 ζ53ζ76+ζ52ζ76 ζ53ζ73+ζ52ζ73 ζ54ζ73+ζ5ζ73 ζ53ζ75+ζ52ζ75 complex faithful

Smallest permutation representation of C7×D5
On 35 points
Generators in S35
(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 30 13 24 20)(2 31 14 25 21)(3 32 8 26 15)(4 33 9 27 16)(5 34 10 28 17)(6 35 11 22 18)(7 29 12 23 19)
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(22 35)(23 29)(24 30)(25 31)(26 32)(27 33)(28 34)

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

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

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

C7×D5 is a maximal subgroup of   C7⋊F5

Matrix representation of C7×D5 in GL2(𝔽29) generated by

 25 0 0 25
,
 23 23 5 0
,
 0 23 24 0
G:=sub<GL(2,GF(29))| [25,0,0,25],[23,5,23,0],[0,24,23,0] >;

C7×D5 in GAP, Magma, Sage, TeX

C_7\times D_5
% in TeX

G:=Group("C7xD5");
// GroupNames label

G:=SmallGroup(70,1);
// by ID

G=gap.SmallGroup(70,1);
# by ID

G:=PCGroup([3,-2,-7,-5,506]);
// Polycyclic

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

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