direct product, metabelian, soluble, monomial, A-group
Aliases: C2×D7⋊A4, D14⋊A4, C23⋊F7, C14⋊(C2×A4), D7⋊(C2×A4), C7⋊A4⋊C22, C7⋊(C22×A4), C22⋊(C2×F7), (C23×D7)⋊3C3, (C22×C14)⋊4C6, (C22×D7)⋊5C6, (C2×C7⋊A4)⋊C2, (C2×C14)⋊3(C2×C6), SmallGroup(336,218)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C14 — C2×D7⋊A4 |
Generators and relations for C2×D7⋊A4
G = < a,b,c,d,e,f | a2=b7=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, fbf-1=b2, cd=dc, ce=ec, fcf-1=bc, fdf-1=de=ed, fef-1=d >
Subgroups: 640 in 78 conjugacy classes, 19 normal (13 characteristic)
C1, C2, C2, C3, C22, C22, C6, C7, C23, C23, A4, C2×C6, D7, D7, C14, C14, C24, C7⋊C3, C2×A4, D14, D14, C2×C14, C2×C14, F7, C2×C7⋊C3, C22×A4, C22×D7, C22×D7, C22×C14, C2×F7, C7⋊A4, C23×D7, D7⋊A4, C2×C7⋊A4, C2×D7⋊A4
Quotients: C1, C2, C3, C22, C6, A4, C2×C6, C2×A4, F7, C22×A4, C2×F7, D7⋊A4, C2×D7⋊A4
Character table of C2×D7⋊A4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 6A | 6B | 6C | 6D | 6E | 6F | 7 | 14A | 14B | 14C | 14D | 14E | 14F | 14G | |
| size | 1 | 1 | 3 | 3 | 7 | 7 | 21 | 21 | 28 | 28 | 28 | 28 | 28 | 28 | 28 | 28 | 6 | 6 | 6 | 6 | 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 | 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 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 2 |
| ρ4 | 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 |
| ρ5 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | ζ32 | ζ3 | ζ6 | ζ32 | ζ65 | ζ6 | ζ65 | ζ3 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 6 |
| ρ6 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | ζ3 | ζ32 | ζ3 | ζ65 | ζ6 | ζ65 | ζ32 | ζ6 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 6 |
| ρ7 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | -1 | ζ32 | ζ3 | ζ32 | ζ6 | ζ65 | ζ6 | ζ3 | ζ65 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 6 |
| ρ8 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | ζ3 | ζ32 | ζ65 | ζ3 | ζ6 | ζ65 | ζ6 | ζ32 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | linear of order 6 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ3 | ζ32 | ζ65 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ32 | ζ3 | ζ6 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ12 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ13 | 3 | -3 | 1 | -1 | -3 | 3 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | -1 | -1 | 1 | 1 | 1 | -3 | -1 | orthogonal lifted from C2×A4 |
| ρ14 | 3 | 3 | -1 | -1 | -3 | -3 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | orthogonal lifted from C2×A4 |
| ρ15 | 3 | 3 | -1 | -1 | 3 | 3 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | -1 | -1 | -1 | -1 | -1 | 3 | -1 | orthogonal lifted from A4 |
| ρ16 | 3 | -3 | 1 | -1 | 3 | -3 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | -1 | -1 | 1 | 1 | 1 | -3 | -1 | orthogonal lifted from C2×A4 |
| ρ17 | 6 | 6 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F7 |
| ρ18 | 6 | -6 | -6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | orthogonal lifted from C2×F7 |
| ρ19 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2ζ76+2ζ7+1 | 2ζ75+2ζ72+1 | 2ζ74+2ζ73+1 | 2ζ76+2ζ7+1 | 2ζ75+2ζ72+1 | -1 | 2ζ74+2ζ73+1 | orthogonal lifted from D7⋊A4 |
| ρ20 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2ζ76+2ζ7+1 | 2ζ75+2ζ72+1 | -2ζ74-2ζ73-1 | -2ζ76-2ζ7-1 | -2ζ75-2ζ72-1 | 1 | 2ζ74+2ζ73+1 | orthogonal faithful |
| ρ21 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2ζ75+2ζ72+1 | 2ζ74+2ζ73+1 | -2ζ76-2ζ7-1 | -2ζ75-2ζ72-1 | -2ζ74-2ζ73-1 | 1 | 2ζ76+2ζ7+1 | orthogonal faithful |
| ρ22 | 6 | -6 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2ζ74+2ζ73+1 | 2ζ76+2ζ7+1 | -2ζ75-2ζ72-1 | -2ζ74-2ζ73-1 | -2ζ76-2ζ7-1 | 1 | 2ζ75+2ζ72+1 | orthogonal faithful |
| ρ23 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2ζ75+2ζ72+1 | 2ζ74+2ζ73+1 | 2ζ76+2ζ7+1 | 2ζ75+2ζ72+1 | 2ζ74+2ζ73+1 | -1 | 2ζ76+2ζ7+1 | orthogonal lifted from D7⋊A4 |
| ρ24 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 2ζ74+2ζ73+1 | 2ζ76+2ζ7+1 | 2ζ75+2ζ72+1 | 2ζ74+2ζ73+1 | 2ζ76+2ζ7+1 | -1 | 2ζ75+2ζ72+1 | orthogonal lifted from D7⋊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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 23)(16 22)(17 28)(18 27)(19 26)(20 25)(21 24)(29 41)(30 40)(31 39)(32 38)(33 37)(34 36)(35 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 32 16)(2 29 18)(3 33 20)(4 30 15)(5 34 17)(6 31 19)(7 35 21)(8 39 23)(9 36 25)(10 40 27)(11 37 22)(12 41 24)(13 38 26)(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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,23)(16,22)(17,28)(18,27)(19,26)(20,25)(21,24)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(35,42), (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), (1,32,16)(2,29,18)(3,33,20)(4,30,15)(5,34,17)(6,31,19)(7,35,21)(8,39,23)(9,36,25)(10,40,27)(11,37,22)(12,41,24)(13,38,26)(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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8)(15,23)(16,22)(17,28)(18,27)(19,26)(20,25)(21,24)(29,41)(30,40)(31,39)(32,38)(33,37)(34,36)(35,42), (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), (1,32,16)(2,29,18)(3,33,20)(4,30,15)(5,34,17)(6,31,19)(7,35,21)(8,39,23)(9,36,25)(10,40,27)(11,37,22)(12,41,24)(13,38,26)(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)], [(1,14),(2,13),(3,12),(4,11),(5,10),(6,9),(7,8),(15,23),(16,22),(17,28),(18,27),(19,26),(20,25),(21,24),(29,41),(30,40),(31,39),(32,38),(33,37),(34,36),(35,42)], [(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)], [(1,32,16),(2,29,18),(3,33,20),(4,30,15),(5,34,17),(6,31,19),(7,35,21),(8,39,23),(9,36,25),(10,40,27),(11,37,22),(12,41,24),(13,38,26),(14,42,28)]])
Matrix representation of C2×D7⋊A4 ►in GL6(𝔽43)
| 42 | 0 | 0 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 0 | 0 | 0 |
| 0 | 0 | 0 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 0 |
| 0 | 0 | 0 | 0 | 0 | 42 |
| 19 | 42 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 24 | 16 | 0 | 0 |
| 0 | 0 | 27 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 24 |
| 0 | 0 | 0 | 0 | 19 | 42 |
| 24 | 1 | 0 | 0 | 0 | 0 |
| 27 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 27 | 0 | 0 |
| 0 | 0 | 1 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 19 |
| 0 | 0 | 0 | 0 | 16 | 16 |
| 42 | 0 | 0 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 0 | 0 | 0 |
| 0 | 0 | 0 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 42 | 0 | 0 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 0 |
| 0 | 0 | 0 | 0 | 0 | 42 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,GF(43))| [42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42],[19,1,0,0,0,0,42,0,0,0,0,0,0,0,24,27,0,0,0,0,16,27,0,0,0,0,0,0,16,19,0,0,0,0,24,42],[24,27,0,0,0,0,1,19,0,0,0,0,0,0,19,1,0,0,0,0,27,24,0,0,0,0,0,0,27,16,0,0,0,0,19,16],[42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[42,0,0,0,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,0,0,0,0,0,0,42],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
C2×D7⋊A4 in GAP, Magma, Sage, TeX
C_2\times D_7\rtimes A_4
% in TeX
G:=Group("C2xD7:A4"); // GroupNames label
G:=SmallGroup(336,218);
// by ID
G=gap.SmallGroup(336,218);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,2,-7,159,286,10373,1745]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^7=c^2=d^2=e^2=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,f*b*f^-1=b^2,c*d=d*c,c*e=e*c,f*c*f^-1=b*c,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations
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