direct product, metabelian, soluble, monomial, A-group
Aliases: C2×C7⋊A4, C14⋊A4, C7⋊2(C2×A4), (C2×C14)⋊7C6, C23⋊2(C7⋊C3), (C22×C14)⋊3C3, C22⋊(C2×C7⋊C3), SmallGroup(168,53)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C14 — C2×C7⋊A4 |
Generators and relations for C2×C7⋊A4
G = < a,b,c,d,e | a2=b7=c2=d2=e3=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b4, ece-1=cd=dc, ede-1=c >
Character table of C2×C7⋊A4
| class | 1 | 2A | 2B | 2C | 3A | 3B | 6A | 6B | 7A | 7B | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | 14J | 14K | 14L | 14M | 14N | |
| size | 1 | 1 | 3 | 3 | 28 | 28 | 28 | 28 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| 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 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | -1 | -1 | 1 | ζ3 | ζ32 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | -1 | -1 | 1 | ζ32 | ζ3 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 6 |
| ρ7 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | -1 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ8 | 3 | -3 | 1 | -1 | 0 | 0 | 0 | 0 | 3 | 3 | -1 | -1 | -1 | -3 | -3 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | orthogonal lifted from C2×A4 |
| ρ9 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -ζ74+ζ72-ζ7 | ζ74-ζ72-ζ7 | ζ76-ζ75-ζ73 | -1+√-7/2 | -1-√-7/2 | -ζ74-ζ72+ζ7 | -ζ74+ζ72-ζ7 | ζ74-ζ72-ζ7 | ζ76-ζ75-ζ73 | -ζ74-ζ72+ζ7 | -ζ76-ζ75+ζ73 | -ζ76+ζ75-ζ73 | -ζ76-ζ75+ζ73 | -ζ76+ζ75-ζ73 | complex lifted from C7⋊A4 |
| ρ10 | 3 | -3 | 1 | -1 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -ζ74+ζ72-ζ7 | ζ74-ζ72-ζ7 | ζ76-ζ75-ζ73 | 1-√-7/2 | 1+√-7/2 | -ζ74-ζ72+ζ7 | ζ74-ζ72+ζ7 | -ζ74+ζ72+ζ7 | -ζ76+ζ75+ζ73 | ζ74+ζ72-ζ7 | ζ76+ζ75-ζ73 | ζ76-ζ75+ζ73 | -ζ76-ζ75+ζ73 | -ζ76+ζ75-ζ73 | complex faithful |
| ρ11 | 3 | -3 | 1 | -1 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -ζ74-ζ72+ζ7 | -ζ74+ζ72-ζ7 | -ζ76-ζ75+ζ73 | 1-√-7/2 | 1+√-7/2 | ζ74-ζ72-ζ7 | ζ74+ζ72-ζ7 | ζ74-ζ72+ζ7 | ζ76+ζ75-ζ73 | -ζ74+ζ72+ζ7 | ζ76-ζ75+ζ73 | -ζ76+ζ75+ζ73 | -ζ76+ζ75-ζ73 | ζ76-ζ75-ζ73 | complex faithful |
| ρ12 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | ζ74-ζ72-ζ7 | -ζ74-ζ72+ζ7 | -ζ76+ζ75-ζ73 | -1+√-7/2 | -1-√-7/2 | -ζ74+ζ72-ζ7 | ζ74-ζ72-ζ7 | -ζ74-ζ72+ζ7 | -ζ76+ζ75-ζ73 | -ζ74+ζ72-ζ7 | ζ76-ζ75-ζ73 | -ζ76-ζ75+ζ73 | ζ76-ζ75-ζ73 | -ζ76-ζ75+ζ73 | complex lifted from C7⋊A4 |
| ρ13 | 3 | -3 | 1 | -1 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -ζ76-ζ75+ζ73 | ζ76-ζ75-ζ73 | -ζ74+ζ72-ζ7 | 1+√-7/2 | 1-√-7/2 | -ζ76+ζ75-ζ73 | ζ76+ζ75-ζ73 | -ζ76+ζ75+ζ73 | ζ74-ζ72+ζ7 | ζ76-ζ75+ζ73 | ζ74+ζ72-ζ7 | -ζ74+ζ72+ζ7 | -ζ74-ζ72+ζ7 | ζ74-ζ72-ζ7 | complex faithful |
| ρ14 | 3 | -3 | 1 | -1 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | ζ74-ζ72-ζ7 | -ζ74-ζ72+ζ7 | -ζ76+ζ75-ζ73 | 1-√-7/2 | 1+√-7/2 | -ζ74+ζ72-ζ7 | -ζ74+ζ72+ζ7 | ζ74+ζ72-ζ7 | ζ76-ζ75+ζ73 | ζ74-ζ72+ζ7 | -ζ76+ζ75+ζ73 | ζ76+ζ75-ζ73 | ζ76-ζ75-ζ73 | -ζ76-ζ75+ζ73 | complex faithful |
| ρ15 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -ζ74-ζ72+ζ7 | -ζ74+ζ72-ζ7 | -ζ76-ζ75+ζ73 | -1+√-7/2 | -1-√-7/2 | ζ74-ζ72-ζ7 | -ζ74-ζ72+ζ7 | -ζ74+ζ72-ζ7 | -ζ76-ζ75+ζ73 | ζ74-ζ72-ζ7 | -ζ76+ζ75-ζ73 | ζ76-ζ75-ζ73 | -ζ76+ζ75-ζ73 | ζ76-ζ75-ζ73 | complex lifted from C7⋊A4 |
| ρ16 | 3 | -3 | 1 | -1 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | ζ76-ζ75-ζ73 | -ζ76+ζ75-ζ73 | ζ74-ζ72-ζ7 | 1+√-7/2 | 1-√-7/2 | -ζ76-ζ75+ζ73 | -ζ76+ζ75+ζ73 | ζ76-ζ75+ζ73 | -ζ74+ζ72+ζ7 | ζ76+ζ75-ζ73 | ζ74-ζ72+ζ7 | ζ74+ζ72-ζ7 | -ζ74+ζ72-ζ7 | -ζ74-ζ72+ζ7 | complex faithful |
| ρ17 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1+√-7/2 | -1+√-7/2 | complex lifted from C7⋊C3 |
| ρ18 | 3 | -3 | -3 | 3 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1-√-7/2 | -1+√-7/2 | 1+√-7/2 | 1-√-7/2 | -1-√-7/2 | 1+√-7/2 | 1+√-7/2 | 1-√-7/2 | 1+√-7/2 | 1-√-7/2 | 1-√-7/2 | -1+√-7/2 | -1+√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ19 | 3 | -3 | 1 | -1 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -ζ76+ζ75-ζ73 | -ζ76-ζ75+ζ73 | -ζ74-ζ72+ζ7 | 1+√-7/2 | 1-√-7/2 | ζ76-ζ75-ζ73 | ζ76-ζ75+ζ73 | ζ76+ζ75-ζ73 | ζ74+ζ72-ζ7 | -ζ76+ζ75+ζ73 | -ζ74+ζ72+ζ7 | ζ74-ζ72+ζ7 | ζ74-ζ72-ζ7 | -ζ74+ζ72-ζ7 | complex faithful |
| ρ20 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | ζ76-ζ75-ζ73 | -ζ76+ζ75-ζ73 | ζ74-ζ72-ζ7 | -1-√-7/2 | -1+√-7/2 | -ζ76-ζ75+ζ73 | ζ76-ζ75-ζ73 | -ζ76+ζ75-ζ73 | ζ74-ζ72-ζ7 | -ζ76-ζ75+ζ73 | -ζ74+ζ72-ζ7 | -ζ74-ζ72+ζ7 | -ζ74+ζ72-ζ7 | -ζ74-ζ72+ζ7 | complex lifted from C7⋊A4 |
| ρ21 | 3 | -3 | -3 | 3 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | 1-√-7/2 | 1+√-7/2 | -1+√-7/2 | 1-√-7/2 | 1-√-7/2 | 1+√-7/2 | 1-√-7/2 | 1+√-7/2 | 1+√-7/2 | -1-√-7/2 | -1-√-7/2 | complex lifted from C2×C7⋊C3 |
| ρ22 | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1+√-7/2 | -1+√-7/2 | -1-√-7/2 | -1+√-7/2 | -1-√-7/2 | -1-√-7/2 | -1-√-7/2 | -1-√-7/2 | complex lifted from C7⋊C3 |
| ρ23 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -ζ76+ζ75-ζ73 | -ζ76-ζ75+ζ73 | -ζ74-ζ72+ζ7 | -1-√-7/2 | -1+√-7/2 | ζ76-ζ75-ζ73 | -ζ76+ζ75-ζ73 | -ζ76-ζ75+ζ73 | -ζ74-ζ72+ζ7 | ζ76-ζ75-ζ73 | ζ74-ζ72-ζ7 | -ζ74+ζ72-ζ7 | ζ74-ζ72-ζ7 | -ζ74+ζ72-ζ7 | complex lifted from C7⋊A4 |
| ρ24 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | -1+√-7/2 | -1-√-7/2 | -ζ76-ζ75+ζ73 | ζ76-ζ75-ζ73 | -ζ74+ζ72-ζ7 | -1-√-7/2 | -1+√-7/2 | -ζ76+ζ75-ζ73 | -ζ76-ζ75+ζ73 | ζ76-ζ75-ζ73 | -ζ74+ζ72-ζ7 | -ζ76+ζ75-ζ73 | -ζ74-ζ72+ζ7 | ζ74-ζ72-ζ7 | -ζ74-ζ72+ζ7 | ζ74-ζ72-ζ7 | complex lifted from C7⋊A4 |
►On 42 points
Generators in S42
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)
(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)(36 37 38 39 40 41 42)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)
(1 30 18)(2 32 15)(3 34 19)(4 29 16)(5 31 20)(6 33 17)(7 35 21)(8 37 25)(9 39 22)(10 41 26)(11 36 23)(12 38 27)(13 40 24)(14 42 28)
G:=sub<Sym(42)| (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (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)(36,37,38,39,40,41,42), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (1,30,18)(2,32,15)(3,34,19)(4,29,16)(5,31,20)(6,33,17)(7,35,21)(8,37,25)(9,39,22)(10,41,26)(11,36,23)(12,38,27)(13,40,24)(14,42,28)>;
G:=Group( (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (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)(36,37,38,39,40,41,42), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42), (1,30,18)(2,32,15)(3,34,19)(4,29,16)(5,31,20)(6,33,17)(7,35,21)(8,37,25)(9,39,22)(10,41,26)(11,36,23)(12,38,27)(13,40,24)(14,42,28) );
G=PermutationGroup([[(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42)], [(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),(36,37,38,39,40,41,42)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42)], [(1,30,18),(2,32,15),(3,34,19),(4,29,16),(5,31,20),(6,33,17),(7,35,21),(8,37,25),(9,39,22),(10,41,26),(11,36,23),(12,38,27),(13,40,24),(14,42,28)]])
C2×C7⋊A4 is a maximal subgroup of
Dic7⋊A4
C2×C7⋊A4 is a maximal quotient of C28.A4
Matrix representation of C2×C7⋊A4 ►in GL3(𝔽43) generated by
| 42 | 0 | 0 |
| 0 | 42 | 0 |
| 0 | 0 | 42 |
| 21 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 35 |
| 42 | 0 | 0 |
| 0 | 42 | 0 |
| 0 | 0 | 1 |
| 42 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 42 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 1 | 0 | 0 |
G:=sub<GL(3,GF(43))| [42,0,0,0,42,0,0,0,42],[21,0,0,0,11,0,0,0,35],[42,0,0,0,42,0,0,0,1],[42,0,0,0,1,0,0,0,42],[0,0,1,1,0,0,0,1,0] >;
C2×C7⋊A4 in GAP, Magma, Sage, TeX
C_2\times C_7\rtimes A_4
% in TeX
G:=Group("C2xC7:A4"); // GroupNames label
G:=SmallGroup(168,53);
// by ID
G=gap.SmallGroup(168,53);
# by ID
G:=PCGroup([5,-2,-3,-2,2,-7,97,188,609]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^7=c^2=d^2=e^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^4,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
Export
Subgroup lattice of C2×C7⋊A4 in TeX
Character table of C2×C7⋊A4 in TeX