direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×Dic7, C14⋊C4, C22.D7, C2.2D14, C14.4C22, C7⋊2(C2×C4), (C2×C14).C2, SmallGroup(56,6)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C2×Dic7 |
Generators and relations for C2×Dic7
G = < a,b,c | a2=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >
Character table of C2×Dic7
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 7A | 7B | 7C | 14A | 14B | 14C | 14D | 14E | 14F | 14G | 14H | 14I | |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 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 | 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 | -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 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | i | -i | -i | i | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | -i | -i | i | i | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | i | i | -i | -i | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | -i | i | i | -i | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ10 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ12 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | -ζ74-ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ76-ζ7 | ζ74+ζ73 | ζ76+ζ7 | -ζ75-ζ72 | orthogonal lifted from D14 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | -ζ76-ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ75-ζ72 | ζ76+ζ7 | ζ75+ζ72 | -ζ74-ζ73 | orthogonal lifted from D14 |
| ρ14 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | -ζ75-ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ74-ζ73 | ζ75+ζ72 | ζ74+ζ73 | -ζ76-ζ7 | orthogonal lifted from D14 |
| ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | ζ75+ζ72 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | ζ74+ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | ζ76+ζ7 | symplectic lifted from Dic7, Schur index 2 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | ζ74+ζ73 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | ζ75+ζ72 | symplectic lifted from Dic7, Schur index 2 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | symplectic lifted from Dic7, Schur index 2 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | symplectic lifted from Dic7, Schur index 2 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | symplectic lifted from Dic7, Schur index 2 |
| ρ20 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | ζ76+ζ7 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | ζ75+ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | ζ74+ζ73 | symplectic lifted from Dic7, Schur index 2 |
►Regular action on 56 points
Generators in S56
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)
(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)(43 44 45 46 47 48 49 50 51 52 53 54 55 56)
(1 29 8 36)(2 42 9 35)(3 41 10 34)(4 40 11 33)(5 39 12 32)(6 38 13 31)(7 37 14 30)(15 50 22 43)(16 49 23 56)(17 48 24 55)(18 47 25 54)(19 46 26 53)(20 45 27 52)(21 44 28 51)
G:=sub<Sym(56)| (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49), (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)(43,44,45,46,47,48,49,50,51,52,53,54,55,56), (1,29,8,36)(2,42,9,35)(3,41,10,34)(4,40,11,33)(5,39,12,32)(6,38,13,31)(7,37,14,30)(15,50,22,43)(16,49,23,56)(17,48,24,55)(18,47,25,54)(19,46,26,53)(20,45,27,52)(21,44,28,51)>;
G:=Group( (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49), (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)(43,44,45,46,47,48,49,50,51,52,53,54,55,56), (1,29,8,36)(2,42,9,35)(3,41,10,34)(4,40,11,33)(5,39,12,32)(6,38,13,31)(7,37,14,30)(15,50,22,43)(16,49,23,56)(17,48,24,55)(18,47,25,54)(19,46,26,53)(20,45,27,52)(21,44,28,51) );
G=PermutationGroup([[(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,43),(37,44),(38,45),(39,46),(40,47),(41,48),(42,49)], [(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),(43,44,45,46,47,48,49,50,51,52,53,54,55,56)], [(1,29,8,36),(2,42,9,35),(3,41,10,34),(4,40,11,33),(5,39,12,32),(6,38,13,31),(7,37,14,30),(15,50,22,43),(16,49,23,56),(17,48,24,55),(18,47,25,54),(19,46,26,53),(20,45,27,52),(21,44,28,51)]])
C2×Dic7 is a maximal subgroup of
Dic7⋊C4 C4⋊Dic7 D14⋊C4 C23.D7 C2×C4×D7 D4⋊2D7
C2×Dic7 is a maximal quotient of C4.Dic7 C4⋊Dic7 C23.D7
Matrix representation of C2×Dic7 ►in GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 28 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 7 |
| 12 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 25 | 8 |
| 0 | 0 | 9 | 4 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[28,0,0,0,0,1,0,0,0,0,0,28,0,0,1,7],[12,0,0,0,0,28,0,0,0,0,25,9,0,0,8,4] >;
C2×Dic7 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_7 % in TeX
G:=Group("C2xDic7"); // GroupNames label
G:=SmallGroup(56,6);
// by ID
G=gap.SmallGroup(56,6);
# by ID
G:=PCGroup([4,-2,-2,-2,-7,16,771]);
// Polycyclic
G:=Group<a,b,c|a^2=b^14=1,c^2=b^7,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export
Subgroup lattice of C2×Dic7 in TeX
Character table of C2×Dic7 in TeX