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G = C7×D5  order 70 = 2·5·7

Direct product of C7 and D5

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

Aliases: C7×D5, C5⋊C14, C35⋊3C2, 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 >

C1
5C2
C5
C7
D5
5C14
C35
C7×D5

Character table of C7×D5

 class 125A5B7A7B7C7D7E7F14A14B14C14D14E14F35A35B35C35D35E35F35G35H35I35J35K35L
 size 1522111111555555222222222222
ρ11111111111111111111111111111    trivial
ρ21-111111111-1-1-1-1-1-1111111111111    linear of order 2
ρ31111ζ72ζ73ζ74ζ75ζ76ζ7ζ76ζ72ζ73ζ74ζ75ζ7ζ75ζ75ζ72ζ76ζ76ζ7ζ73ζ73ζ7ζ74ζ74ζ72    linear of order 7
ρ41111ζ75ζ74ζ73ζ72ζ7ζ76ζ7ζ75ζ74ζ73ζ72ζ76ζ72ζ72ζ75ζ7ζ7ζ76ζ74ζ74ζ76ζ73ζ73ζ75    linear of order 7
ρ51-111ζ7ζ75ζ72ζ76ζ73ζ74-ζ73-ζ7-ζ75-ζ72-ζ76-ζ74ζ76ζ76ζ7ζ73ζ73ζ74ζ75ζ75ζ74ζ72ζ72ζ7    linear of order 14
ρ61-111ζ72ζ73ζ74ζ75ζ76ζ7-ζ76-ζ72-ζ73-ζ74-ζ75-ζ7ζ75ζ75ζ72ζ76ζ76ζ7ζ73ζ73ζ7ζ74ζ74ζ72    linear of order 14
ρ71-111ζ73ζ7ζ76ζ74ζ72ζ75-ζ72-ζ73-ζ7-ζ76-ζ74-ζ75ζ74ζ74ζ73ζ72ζ72ζ75ζ7ζ7ζ75ζ76ζ76ζ73    linear of order 14
ρ81111ζ7ζ75ζ72ζ76ζ73ζ74ζ73ζ7ζ75ζ72ζ76ζ74ζ76ζ76ζ7ζ73ζ73ζ74ζ75ζ75ζ74ζ72ζ72ζ7    linear of order 7
ρ91-111ζ74ζ76ζ7ζ73ζ75ζ72-ζ75-ζ74-ζ76-ζ7-ζ73-ζ72ζ73ζ73ζ74ζ75ζ75ζ72ζ76ζ76ζ72ζ7ζ7ζ74    linear of order 14
ρ101111ζ76ζ72ζ75ζ7ζ74ζ73ζ74ζ76ζ72ζ75ζ7ζ73ζ7ζ7ζ76ζ74ζ74ζ73ζ72ζ72ζ73ζ75ζ75ζ76    linear of order 7
ρ111-111ζ76ζ72ζ75ζ7ζ74ζ73-ζ74-ζ76-ζ72-ζ75-ζ7-ζ73ζ7ζ7ζ76ζ74ζ74ζ73ζ72ζ72ζ73ζ75ζ75ζ76    linear of order 14
ρ121-111ζ75ζ74ζ73ζ72ζ7ζ76-ζ7-ζ75-ζ74-ζ73-ζ72-ζ76ζ72ζ72ζ75ζ7ζ7ζ76ζ74ζ74ζ76ζ73ζ73ζ75    linear of order 14
ρ131111ζ73ζ7ζ76ζ74ζ72ζ75ζ72ζ73ζ7ζ76ζ74ζ75ζ74ζ74ζ73ζ72ζ72ζ75ζ7ζ7ζ75ζ76ζ76ζ73    linear of order 7
ρ141111ζ74ζ76ζ7ζ73ζ75ζ72ζ75ζ74ζ76ζ7ζ73ζ72ζ73ζ73ζ74ζ75ζ75ζ72ζ76ζ76ζ72ζ7ζ7ζ74    linear of order 7
ρ1520-1-√5/2-1+√5/2222222000000-1+√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
ρ1620-1+√5/2-1-√5/2222222000000-1-√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
ρ1720-1+√5/2-1-√5/22ζ762ζ722ζ752ζ72ζ742ζ73000000ζ53ζ7+ζ52ζ7ζ54ζ7+ζ5ζ7ζ54ζ76+ζ5ζ76ζ53ζ74+ζ52ζ74ζ54ζ74+ζ5ζ74ζ54ζ73+ζ5ζ73ζ53ζ72+ζ52ζ72ζ54ζ72+ζ5ζ72ζ53ζ73+ζ52ζ73ζ53ζ75+ζ52ζ75ζ54ζ75+ζ5ζ75ζ53ζ76+ζ52ζ76    complex faithful
ρ1820-1-√5/2-1+√5/22ζ722ζ732ζ742ζ752ζ762ζ7000000ζ54ζ75+ζ5ζ75ζ53ζ75+ζ52ζ75ζ53ζ72+ζ52ζ72ζ54ζ76+ζ5ζ76ζ53ζ76+ζ52ζ76ζ53ζ7+ζ52ζ7ζ54ζ73+ζ5ζ73ζ53ζ73+ζ52ζ73ζ54ζ7+ζ5ζ7ζ54ζ74+ζ5ζ74ζ53ζ74+ζ52ζ74ζ54ζ72+ζ5ζ72    complex faithful
ρ1920-1-√5/2-1+√5/22ζ752ζ742ζ732ζ722ζ72ζ76000000ζ54ζ72+ζ5ζ72ζ53ζ72+ζ52ζ72ζ53ζ75+ζ52ζ75ζ54ζ7+ζ5ζ7ζ53ζ7+ζ52ζ7ζ53ζ76+ζ52ζ76ζ54ζ74+ζ5ζ74ζ53ζ74+ζ52ζ74ζ54ζ76+ζ5ζ76ζ54ζ73+ζ5ζ73ζ53ζ73+ζ52ζ73ζ54ζ75+ζ5ζ75    complex faithful
ρ2020-1+√5/2-1-√5/22ζ742ζ762ζ72ζ732ζ752ζ72000000ζ53ζ73+ζ52ζ73ζ54ζ73+ζ5ζ73ζ54ζ74+ζ5ζ74ζ53ζ75+ζ52ζ75ζ54ζ75+ζ5ζ75ζ54ζ72+ζ5ζ72ζ53ζ76+ζ52ζ76ζ54ζ76+ζ5ζ76ζ53ζ72+ζ52ζ72ζ53ζ7+ζ52ζ7ζ54ζ7+ζ5ζ7ζ53ζ74+ζ52ζ74    complex faithful
ρ2120-1+√5/2-1-√5/22ζ732ζ72ζ762ζ742ζ722ζ75000000ζ53ζ74+ζ52ζ74ζ54ζ74+ζ5ζ74ζ54ζ73+ζ5ζ73ζ53ζ72+ζ52ζ72ζ54ζ72+ζ5ζ72ζ54ζ75+ζ5ζ75ζ53ζ7+ζ52ζ7ζ54ζ7+ζ5ζ7ζ53ζ75+ζ52ζ75ζ53ζ76+ζ52ζ76ζ54ζ76+ζ5ζ76ζ53ζ73+ζ52ζ73    complex faithful
ρ2220-1-√5/2-1+√5/22ζ72ζ752ζ722ζ762ζ732ζ74000000ζ54ζ76+ζ5ζ76ζ53ζ76+ζ52ζ76ζ53ζ7+ζ52ζ7ζ54ζ73+ζ5ζ73ζ53ζ73+ζ52ζ73ζ53ζ74+ζ52ζ74ζ54ζ75+ζ5ζ75ζ53ζ75+ζ52ζ75ζ54ζ74+ζ5ζ74ζ54ζ72+ζ5ζ72ζ53ζ72+ζ52ζ72ζ54ζ7+ζ5ζ7    complex faithful
ρ2320-1-√5/2-1+√5/22ζ742ζ762ζ72ζ732ζ752ζ72000000ζ54ζ73+ζ5ζ73ζ53ζ73+ζ52ζ73ζ53ζ74+ζ52ζ74ζ54ζ75+ζ5ζ75ζ53ζ75+ζ52ζ75ζ53ζ72+ζ52ζ72ζ54ζ76+ζ5ζ76ζ53ζ76+ζ52ζ76ζ54ζ72+ζ5ζ72ζ54ζ7+ζ5ζ7ζ53ζ7+ζ52ζ7ζ54ζ74+ζ5ζ74    complex faithful
ρ2420-1+√5/2-1-√5/22ζ72ζ752ζ722ζ762ζ732ζ74000000ζ53ζ76+ζ52ζ76ζ54ζ76+ζ5ζ76ζ54ζ7+ζ5ζ7ζ53ζ73+ζ52ζ73ζ54ζ73+ζ5ζ73ζ54ζ74+ζ5ζ74ζ53ζ75+ζ52ζ75ζ54ζ75+ζ5ζ75ζ53ζ74+ζ52ζ74ζ53ζ72+ζ52ζ72ζ54ζ72+ζ5ζ72ζ53ζ7+ζ52ζ7    complex faithful
ρ2520-1-√5/2-1+√5/22ζ762ζ722ζ752ζ72ζ742ζ73000000ζ54ζ7+ζ5ζ7ζ53ζ7+ζ52ζ7ζ53ζ76+ζ52ζ76ζ54ζ74+ζ5ζ74ζ53ζ74+ζ52ζ74ζ53ζ73+ζ52ζ73ζ54ζ72+ζ5ζ72ζ53ζ72+ζ52ζ72ζ54ζ73+ζ5ζ73ζ54ζ75+ζ5ζ75ζ53ζ75+ζ52ζ75ζ54ζ76+ζ5ζ76    complex faithful
ρ2620-1+√5/2-1-√5/22ζ722ζ732ζ742ζ752ζ762ζ7000000ζ53ζ75+ζ52ζ75ζ54ζ75+ζ5ζ75ζ54ζ72+ζ5ζ72ζ53ζ76+ζ52ζ76ζ54ζ76+ζ5ζ76ζ54ζ7+ζ5ζ7ζ53ζ73+ζ52ζ73ζ54ζ73+ζ5ζ73ζ53ζ7+ζ52ζ7ζ53ζ74+ζ52ζ74ζ54ζ74+ζ5ζ74ζ53ζ72+ζ52ζ72    complex faithful
ρ2720-1-√5/2-1+√5/22ζ732ζ72ζ762ζ742ζ722ζ75000000ζ54ζ74+ζ5ζ74ζ53ζ74+ζ52ζ74ζ53ζ73+ζ52ζ73ζ54ζ72+ζ5ζ72ζ53ζ72+ζ52ζ72ζ53ζ75+ζ52ζ75ζ54ζ7+ζ5ζ7ζ53ζ7+ζ52ζ7ζ54ζ75+ζ5ζ75ζ54ζ76+ζ5ζ76ζ53ζ76+ζ52ζ76ζ54ζ73+ζ5ζ73    complex faithful
ρ2820-1+√5/2-1-√5/22ζ752ζ742ζ732ζ722ζ72ζ76000000ζ53ζ72+ζ52ζ72ζ54ζ72+ζ5ζ72ζ54ζ75+ζ5ζ75ζ53ζ7+ζ52ζ7ζ54ζ7+ζ5ζ7ζ54ζ76+ζ5ζ76ζ53ζ74+ζ52ζ74ζ54ζ74+ζ5ζ74ζ53ζ76+ζ52ζ76ζ53ζ73+ζ52ζ73ζ54ζ73+ζ5ζ73ζ53ζ75+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

250
025
,
2323
50
,
023
240
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
 

Export

Subgroup lattice of C7×D5 in TeX
Character table of C7×D5 in TeX

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