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## G = C6×D12order 144 = 24·32

### Direct product of C6 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C6×D12
 Chief series C1 — C3 — C6 — C3×C6 — S3×C6 — S3×C2×C6 — C6×D12
 Lower central C3 — C6 — C6×D12
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C6×D12
G = < a,b,c | a6=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 264 in 116 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×4], C3 [×2], C3, C4 [×2], C22, C22 [×8], S3 [×4], C6 [×2], C6 [×4], C6 [×7], C2×C4, D4 [×4], C23 [×2], C32, C12 [×4], C12 [×2], D6 [×4], D6 [×4], C2×C6 [×2], C2×C6 [×9], C2×D4, C3×S3 [×4], C3×C6, C3×C6 [×2], D12 [×4], C2×C12 [×2], C2×C12, C3×D4 [×4], C22×S3 [×2], C22×C6 [×2], C3×C12 [×2], S3×C6 [×4], S3×C6 [×4], C62, C2×D12, C6×D4, C3×D12 [×4], C6×C12, S3×C2×C6 [×2], C6×D12
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C3×S3, D12 [×2], C3×D4 [×2], C22×S3, C22×C6, S3×C6 [×3], C2×D12, C6×D4, C3×D12 [×2], S3×C2×C6, C6×D12

Smallest permutation representation of C6×D12
On 48 points
Generators in S48
(1 35 5 27 9 31)(2 36 6 28 10 32)(3 25 7 29 11 33)(4 26 8 30 12 34)(13 48 21 44 17 40)(14 37 22 45 18 41)(15 38 23 46 19 42)(16 39 24 47 20 43)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(12 24)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 48)(34 47)(35 46)(36 45)

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

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

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

54 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 6A ··· 6F 6G ··· 6O 6P ··· 6W 12A ··· 12P order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 6 6 6 6 1 1 2 2 2 2 2 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2

54 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 D6 C3×S3 D12 C3×D4 S3×C6 S3×C6 C3×D12 kernel C6×D12 C3×D12 C6×C12 S3×C2×C6 C2×D12 D12 C2×C12 C22×S3 C2×C12 C3×C6 C12 C2×C6 C2×C4 C6 C6 C4 C22 C2 # reps 1 4 1 2 2 8 2 4 1 2 2 1 2 4 4 4 2 8

Matrix representation of C6×D12 in GL3(𝔽13) generated by

 4 0 0 0 3 0 0 0 3
,
 12 0 0 0 6 0 0 0 11
,
 1 0 0 0 0 11 0 6 0
G:=sub<GL(3,GF(13))| [4,0,0,0,3,0,0,0,3],[12,0,0,0,6,0,0,0,11],[1,0,0,0,0,6,0,11,0] >;

C6×D12 in GAP, Magma, Sage, TeX

C_6\times D_{12}
% in TeX

G:=Group("C6xD12");
// GroupNames label

G:=SmallGroup(144,160);
// by ID

G=gap.SmallGroup(144,160);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-3,506,122,3461]);
// Polycyclic

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

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