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## G = C2×Dic6order 48 = 24·3

### Direct product of C2 and Dic6

Aliases: C2×Dic6, C6⋊Q8, C4.11D6, C6.1C23, C22.8D6, C12.11C22, Dic3.1C22, C31(C2×Q8), (C2×C4).4S3, (C2×C12).3C2, C2.3(C22×S3), (C2×C6).8C22, (C2×Dic3).3C2, SmallGroup(48,34)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×Dic6
 Chief series C1 — C3 — C6 — Dic3 — C2×Dic3 — C2×Dic6
 Lower central C3 — C6 — C2×Dic6
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×Dic6
G = < a,b,c | a2=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

Character table of C2×Dic6

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 12A 12B 12C 12D size 1 1 1 1 2 2 2 6 6 6 6 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 trivial ρ2 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 linear of order 2 ρ4 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 -1 -1 -1 1 1 1 1 1 1 1 linear of order 2 ρ6 1 -1 -1 1 1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 -1 1 linear of order 2 ρ7 1 1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ8 1 -1 -1 1 1 -1 1 1 1 -1 -1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ9 2 -2 -2 2 -1 -2 2 0 0 0 0 1 -1 1 -1 1 -1 1 orthogonal lifted from D6 ρ10 2 2 2 2 -1 2 2 0 0 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ11 2 -2 -2 2 -1 2 -2 0 0 0 0 1 -1 1 1 -1 1 -1 orthogonal lifted from D6 ρ12 2 2 2 2 -1 -2 -2 0 0 0 0 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ13 2 -2 2 -2 2 0 0 0 0 0 0 2 -2 -2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ14 2 2 -2 -2 2 0 0 0 0 0 0 -2 -2 2 0 0 0 0 symplectic lifted from Q8, Schur index 2 ρ15 2 -2 2 -2 -1 0 0 0 0 0 0 -1 1 1 √3 √3 -√3 -√3 symplectic lifted from Dic6, Schur index 2 ρ16 2 2 -2 -2 -1 0 0 0 0 0 0 1 1 -1 √3 -√3 -√3 √3 symplectic lifted from Dic6, Schur index 2 ρ17 2 -2 2 -2 -1 0 0 0 0 0 0 -1 1 1 -√3 -√3 √3 √3 symplectic lifted from Dic6, Schur index 2 ρ18 2 2 -2 -2 -1 0 0 0 0 0 0 1 1 -1 -√3 √3 √3 -√3 symplectic lifted from Dic6, Schur index 2

Smallest permutation representation of C2×Dic6
Regular action on 48 points
Generators in S48
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)
(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)
(1 31 7 25)(2 30 8 36)(3 29 9 35)(4 28 10 34)(5 27 11 33)(6 26 12 32)(13 40 19 46)(14 39 20 45)(15 38 21 44)(16 37 22 43)(17 48 23 42)(18 47 24 41)

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

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

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

C2×Dic6 is a maximal subgroup of
C6.SD16  C2.Dic12  C12.47D4  C122Q8  C427S3  Dic3.D4  C23.11D6  Dic6⋊C4  C12⋊Q8  D6⋊Q8  C4.D12  C8.D6  C12.48D4  C23.12D6  Dic3⋊Q8  Q8.14D6  C2×S3×Q8  Q8○D12  Q8⋊Dic6
C2×Dic6 is a maximal quotient of
C122Q8  C12.6Q8  Dic3.D4  C12⋊Q8  C4.Dic6  C12.48D4

Matrix representation of C2×Dic6 in GL5(𝔽13)

 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 4 0 0 0 0 0 10 0 0 0 0 0 12 5 0 0 0 10 1
,
 12 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 6 2 0 0 0 1 7

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,10,0,0,0,0,0,12,10,0,0,0,5,1],[12,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,6,1,0,0,0,2,7] >;

C2×Dic6 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_6
% in TeX

G:=Group("C2xDic6");
// GroupNames label

G:=SmallGroup(48,34);
// by ID

G=gap.SmallGroup(48,34);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-3,40,182,42,804]);
// Polycyclic

G:=Group<a,b,c|a^2=b^12=1,c^2=b^6,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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