direct product, metacyclic, supersoluble, monomial, Z-group
Aliases: D5×C7⋊C3, C35⋊3C6, (C7×D5)⋊C3, C7⋊2(C3×D5), C5⋊(C2×C7⋊C3), (C5×C7⋊C3)⋊3C2, SmallGroup(210,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C35 — D5×C7⋊C3 |
Generators and relations for D5×C7⋊C3
G = < a,b,c,d | a5=b2=c7=d3=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c4 >
Character table of D5×C7⋊C3
| class | 1 | 2 | 3A | 3B | 5A | 5B | 6A | 6B | 7A | 7B | 14A | 14B | 15A | 15B | 15C | 15D | 35A | 35B | 35C | 35D | |
| size | 1 | 5 | 7 | 7 | 2 | 2 | 35 | 35 | 3 | 3 | 15 | 15 | 14 | 14 | 14 | 14 | 6 | 6 | 6 | 6 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | ζ6 | ζ65 | 1 | 1 | -1 | -1 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ5 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ6 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | ζ65 | ζ6 | 1 | 1 | -1 | -1 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 2 | 0 | 2 | 2 | -1+√5/2 | -1-√5/2 | 0 | 0 | 2 | 2 | 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 | orthogonal lifted from D5 |
| ρ8 | 2 | 0 | 2 | 2 | -1-√5/2 | -1+√5/2 | 0 | 0 | 2 | 2 | 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 | orthogonal lifted from D5 |
| ρ9 | 2 | 0 | -1-√-3 | -1+√-3 | -1-√5/2 | -1+√5/2 | 0 | 0 | 2 | 2 | 0 | 0 | ζ3ζ54+ζ3ζ5 | ζ3ζ53+ζ3ζ52 | ζ32ζ54+ζ32ζ5 | ζ32ζ53+ζ32ζ52 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | complex lifted from C3×D5 |
| ρ10 | 2 | 0 | -1+√-3 | -1-√-3 | -1+√5/2 | -1-√5/2 | 0 | 0 | 2 | 2 | 0 | 0 | ζ32ζ53+ζ32ζ52 | ζ32ζ54+ζ32ζ5 | ζ3ζ53+ζ3ζ52 | ζ3ζ54+ζ3ζ5 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | complex lifted from C3×D5 |
| ρ11 | 2 | 0 | -1+√-3 | -1-√-3 | -1-√5/2 | -1+√5/2 | 0 | 0 | 2 | 2 | 0 | 0 | ζ32ζ54+ζ32ζ5 | ζ32ζ53+ζ32ζ52 | ζ3ζ54+ζ3ζ5 | ζ3ζ53+ζ3ζ52 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | complex lifted from C3×D5 |
| ρ12 | 2 | 0 | -1-√-3 | -1+√-3 | -1+√5/2 | -1-√5/2 | 0 | 0 | 2 | 2 | 0 | 0 | ζ3ζ53+ζ3ζ52 | ζ3ζ54+ζ3ζ5 | ζ32ζ53+ζ32ζ52 | ζ32ζ54+ζ32ζ5 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | complex lifted from C3×D5 |
| ρ13 | 3 | 3 | 0 | 0 | 3 | 3 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | complex lifted from C7⋊C3 |
| ρ14 | 3 | -3 | 0 | 0 | 3 | 3 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | 1+√-7/2 | 1-√-7/2 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ15 | 3 | 3 | 0 | 0 | 3 | 3 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | complex lifted from C7⋊C3 |
| ρ16 | 3 | -3 | 0 | 0 | 3 | 3 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | 1-√-7/2 | 1+√-7/2 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ17 | 6 | 0 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | -1-√-7 | -1+√-7 | 0 | 0 | 0 | 0 | 0 | 0 | ζ54ζ76+ζ54ζ75+ζ54ζ73+ζ5ζ76+ζ5ζ75+ζ5ζ73 | ζ53ζ76+ζ53ζ75+ζ53ζ73+ζ52ζ76+ζ52ζ75+ζ52ζ73 | ζ53ζ74+ζ53ζ72+ζ53ζ7+ζ52ζ74+ζ52ζ72+ζ52ζ7 | ζ54ζ74+ζ54ζ72+ζ54ζ7+ζ5ζ74+ζ5ζ72+ζ5ζ7 | complex faithful |
| ρ18 | 6 | 0 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 0 | -1+√-7 | -1-√-7 | 0 | 0 | 0 | 0 | 0 | 0 | ζ54ζ74+ζ54ζ72+ζ54ζ7+ζ5ζ74+ζ5ζ72+ζ5ζ7 | ζ53ζ74+ζ53ζ72+ζ53ζ7+ζ52ζ74+ζ52ζ72+ζ52ζ7 | ζ53ζ76+ζ53ζ75+ζ53ζ73+ζ52ζ76+ζ52ζ75+ζ52ζ73 | ζ54ζ76+ζ54ζ75+ζ54ζ73+ζ5ζ76+ζ5ζ75+ζ5ζ73 | complex faithful |
| ρ19 | 6 | 0 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | -1-√-7 | -1+√-7 | 0 | 0 | 0 | 0 | 0 | 0 | ζ53ζ76+ζ53ζ75+ζ53ζ73+ζ52ζ76+ζ52ζ75+ζ52ζ73 | ζ54ζ76+ζ54ζ75+ζ54ζ73+ζ5ζ76+ζ5ζ75+ζ5ζ73 | ζ54ζ74+ζ54ζ72+ζ54ζ7+ζ5ζ74+ζ5ζ72+ζ5ζ7 | ζ53ζ74+ζ53ζ72+ζ53ζ7+ζ52ζ74+ζ52ζ72+ζ52ζ7 | complex faithful |
| ρ20 | 6 | 0 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 0 | -1+√-7 | -1-√-7 | 0 | 0 | 0 | 0 | 0 | 0 | ζ53ζ74+ζ53ζ72+ζ53ζ7+ζ52ζ74+ζ52ζ72+ζ52ζ7 | ζ54ζ74+ζ54ζ72+ζ54ζ7+ζ5ζ74+ζ5ζ72+ζ5ζ7 | ζ54ζ76+ζ54ζ75+ζ54ζ73+ζ5ζ76+ζ5ζ75+ζ5ζ73 | ζ53ζ76+ζ53ζ75+ζ53ζ73+ζ52ζ76+ζ52ζ75+ζ52ζ73 | complex faithful |
►On 35 points
Generators in S35
(1 29 22 15 8)(2 30 23 16 9)(3 31 24 17 10)(4 32 25 18 11)(5 33 26 19 12)(6 34 27 20 13)(7 35 28 21 14)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 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)
(2 3 5)(4 7 6)(9 10 12)(11 14 13)(16 17 19)(18 21 20)(23 24 26)(25 28 27)(30 31 33)(32 35 34)
G:=sub<Sym(35)| (1,29,22,15,8)(2,30,23,16,9)(3,31,24,17,10)(4,32,25,18,11)(5,33,26,19,12)(6,34,27,20,13)(7,35,28,21,14), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35), (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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)(30,31,33)(32,35,34)>;
G:=Group( (1,29,22,15,8)(2,30,23,16,9)(3,31,24,17,10)(4,32,25,18,11)(5,33,26,19,12)(6,34,27,20,13)(7,35,28,21,14), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35), (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), (2,3,5)(4,7,6)(9,10,12)(11,14,13)(16,17,19)(18,21,20)(23,24,26)(25,28,27)(30,31,33)(32,35,34) );
G=PermutationGroup([[(1,29,22,15,8),(2,30,23,16,9),(3,31,24,17,10),(4,32,25,18,11),(5,33,26,19,12),(6,34,27,20,13),(7,35,28,21,14)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35)], [(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)], [(2,3,5),(4,7,6),(9,10,12),(11,14,13),(16,17,19),(18,21,20),(23,24,26),(25,28,27),(30,31,33),(32,35,34)]])
D5×C7⋊C3 is a maximal subgroup of
C35⋊C12
Matrix representation of D5×C7⋊C3 ►in GL5(𝔽211)
| 1 | 6 | 0 | 0 | 0 |
| 170 | 177 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 34 | 6 | 0 | 0 | 0 |
| 124 | 177 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 210 | 190 | 1 |
| 0 | 0 | 0 | 190 | 1 |
| 0 | 0 | 210 | 191 | 1 |
| 14 | 0 | 0 | 0 | 0 |
| 0 | 14 | 0 | 0 | 0 |
| 0 | 0 | 191 | 1 | 21 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 | 20 |
G:=sub<GL(5,GF(211))| [1,170,0,0,0,6,177,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[34,124,0,0,0,6,177,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,210,0,210,0,0,190,190,191,0,0,1,1,1],[14,0,0,0,0,0,14,0,0,0,0,0,191,1,1,0,0,1,0,1,0,0,21,0,20] >;
D5×C7⋊C3 in GAP, Magma, Sage, TeX
D_5\times C_7\rtimes C_3
% in TeX
G:=Group("D5xC7:C3"); // GroupNames label
G:=SmallGroup(210,2);
// by ID
G=gap.SmallGroup(210,2);
# by ID
G:=PCGroup([4,-2,-3,-5,-7,290,487]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^7=d^3=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^4>;
// generators/relations
Export
Subgroup lattice of D5×C7⋊C3 in TeX
Character table of D5×C7⋊C3 in TeX