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G = C2×Dic7order 56 = 23·7

Direct product of C2 and Dic7

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

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

C1C7 — C2×Dic7
C1C7C14Dic7 — C2×Dic7
C7 — C2×Dic7
C1C22

Generators and relations for C2×Dic7
 G = < a,b,c | a2=b14=1, c2=b7, ab=ba, ac=ca, cbc-1=b-1 >

7C4
7C4
7C2×C4

Character table of C2×Dic7

 class 12A2B2C4A4B4C4D7A7B7C14A14B14C14D14E14F14G14H14I
 size 11117777222222222222
ρ111111111111111111111    trivial
ρ211-1-1-11-111111-1-1-1-1-111-1    linear of order 2
ρ311-1-11-11-11111-1-1-1-1-111-1    linear of order 2
ρ41111-1-1-1-1111111111111    linear of order 2
ρ51-1-11i-i-ii111-1-1111-1-1-1-1    linear of order 4
ρ61-11-1-i-iii111-11-1-1-11-1-11    linear of order 4
ρ71-11-1ii-i-i111-11-1-1-11-1-11    linear of order 4
ρ81-1-11-iii-i111-1-1111-1-1-1-1    linear of order 4
ρ922220000ζ7572ζ7473ζ767ζ7473ζ767ζ767ζ7572ζ7473ζ7572ζ767ζ7572ζ7473    orthogonal lifted from D7
ρ1022220000ζ767ζ7572ζ7473ζ7572ζ7473ζ7473ζ767ζ7572ζ767ζ7473ζ767ζ7572    orthogonal lifted from D7
ρ1122220000ζ7473ζ767ζ7572ζ767ζ7572ζ7572ζ7473ζ767ζ7473ζ7572ζ7473ζ767    orthogonal lifted from D7
ρ1222-2-20000ζ767ζ7572ζ7473ζ7572747374737677572767ζ7473ζ7677572    orthogonal lifted from D14
ρ1322-2-20000ζ7572ζ7473ζ767ζ7473767767757274737572ζ767ζ75727473    orthogonal lifted from D14
ρ1422-2-20000ζ7473ζ767ζ7572ζ7677572757274737677473ζ7572ζ7473767    orthogonal lifted from D14
ρ152-22-20000ζ7473ζ767ζ7572767ζ757275727473767ζ747375727473ζ767    symplectic lifted from Dic7, Schur index 2
ρ162-22-20000ζ767ζ7572ζ74737572ζ747374737677572ζ7677473767ζ7572    symplectic lifted from Dic7, Schur index 2
ρ172-2-220000ζ767ζ7572ζ747375727473ζ7473ζ767ζ757276774737677572    symplectic lifted from Dic7, Schur index 2
ρ182-2-220000ζ7572ζ7473ζ7677473767ζ767ζ7572ζ7473757276775727473    symplectic lifted from Dic7, Schur index 2
ρ192-2-220000ζ7473ζ767ζ75727677572ζ7572ζ7473ζ767747375727473767    symplectic lifted from Dic7, Schur index 2
ρ202-22-20000ζ7572ζ7473ζ7677473ζ76776775727473ζ75727677572ζ7473    symplectic lifted from Dic7, Schur index 2

Smallest permutation representation of C2×Dic7
Regular action on 56 points
Generators in S56
(1 27)(2 28)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(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 36 8 29)(2 35 9 42)(3 34 10 41)(4 33 11 40)(5 32 12 39)(6 31 13 38)(7 30 14 37)(15 55 22 48)(16 54 23 47)(17 53 24 46)(18 52 25 45)(19 51 26 44)(20 50 27 43)(21 49 28 56)

G:=sub<Sym(56)| (1,27)(2,28)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(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), (1,36,8,29)(2,35,9,42)(3,34,10,41)(4,33,11,40)(5,32,12,39)(6,31,13,38)(7,30,14,37)(15,55,22,48)(16,54,23,47)(17,53,24,46)(18,52,25,45)(19,51,26,44)(20,50,27,43)(21,49,28,56)>;

G:=Group( (1,27)(2,28)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(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), (1,36,8,29)(2,35,9,42)(3,34,10,41)(4,33,11,40)(5,32,12,39)(6,31,13,38)(7,30,14,37)(15,55,22,48)(16,54,23,47)(17,53,24,46)(18,52,25,45)(19,51,26,44)(20,50,27,43)(21,49,28,56) );

G=PermutationGroup([(1,27),(2,28),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(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)], [(1,36,8,29),(2,35,9,42),(3,34,10,41),(4,33,11,40),(5,32,12,39),(6,31,13,38),(7,30,14,37),(15,55,22,48),(16,54,23,47),(17,53,24,46),(18,52,25,45),(19,51,26,44),(20,50,27,43),(21,49,28,56)])

Matrix representation of C2×Dic7 in GL4(𝔽29) generated by

28000
02800
0010
0001
,
28000
0100
0001
00287
,
12000
02800
00258
0094
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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