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## G = C2×Dic7order 56 = 23·7

### Direct product of C2 and Dic7

Aliases: C2×Dic7, C14⋊C4, C22.D7, C2.2D14, C14.4C22, C72(C2×C4), (C2×C14).C2, SmallGroup(56,6)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C2×Dic7
 Chief series C1 — C7 — C14 — Dic7 — C2×Dic7
 Lower central C7 — C2×Dic7
 Upper central C1 — C22

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

Smallest permutation representation of C2×Dic7
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  D42D7
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

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