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## G = Dic3×C12order 144 = 24·32

### Direct product of C12 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — Dic3×C12
 Chief series C1 — C3 — C6 — C2×C6 — C62 — C6×Dic3 — Dic3×C12
 Lower central C3 — Dic3×C12
 Upper central C1 — C2×C12

Generators and relations for Dic3×C12
G = < a,b,c | a12=b6=1, c2=b3, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 104 in 68 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4 [×2], C4 [×4], C22, C6 [×2], C6 [×4], C6 [×3], C2×C4, C2×C4 [×2], C32, Dic3 [×4], C12 [×4], C12 [×6], C2×C6 [×2], C2×C6, C42, C3×C6, C3×C6 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×Dic3 [×4], C3×C12 [×2], C62, C4×Dic3, C4×C12, C6×Dic3 [×2], C6×C12, Dic3×C12
Quotients: C1, C2 [×3], C3, C4 [×6], C22, S3, C6 [×3], C2×C4 [×3], Dic3 [×2], C12 [×6], D6, C2×C6, C42, C3×S3, C4×S3 [×2], C2×Dic3, C2×C12 [×3], C3×Dic3 [×2], S3×C6, C4×Dic3, C4×C12, S3×C12 [×2], C6×Dic3, Dic3×C12

Smallest permutation representation of Dic3×C12
On 48 points
Generators in S48
(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 17 5 21 9 13)(2 18 6 22 10 14)(3 19 7 23 11 15)(4 20 8 24 12 16)(25 39 33 47 29 43)(26 40 34 48 30 44)(27 41 35 37 31 45)(28 42 36 38 32 46)
(1 40 21 30)(2 41 22 31)(3 42 23 32)(4 43 24 33)(5 44 13 34)(6 45 14 35)(7 46 15 36)(8 47 16 25)(9 48 17 26)(10 37 18 27)(11 38 19 28)(12 39 20 29)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E ··· 4L 6A ··· 6F 6G ··· 6O 12A ··· 12H 12I ··· 12T 12U ··· 12AJ order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 1 1 2 2 2 1 1 1 1 3 ··· 3 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 3 ··· 3

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + - + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 S3 Dic3 D6 C3×S3 C4×S3 C3×Dic3 S3×C6 S3×C12 kernel Dic3×C12 C6×Dic3 C6×C12 C4×Dic3 C3×Dic3 C3×C12 C2×Dic3 C2×C12 Dic3 C12 C2×C12 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 2 1 2 8 4 4 2 16 8 1 2 1 2 4 4 2 8

Matrix representation of Dic3×C12 in GL3(𝔽13) generated by

 9 0 0 0 2 0 0 0 2
,
 12 0 0 0 9 2 0 0 3
,
 8 0 0 0 12 0 0 3 1
G:=sub<GL(3,GF(13))| [9,0,0,0,2,0,0,0,2],[12,0,0,0,9,0,0,2,3],[8,0,0,0,12,3,0,0,1] >;

Dic3×C12 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_{12}
% in TeX

G:=Group("Dic3xC12");
// GroupNames label

G:=SmallGroup(144,76);
// by ID

G=gap.SmallGroup(144,76);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,72,151,3461]);
// Polycyclic

G:=Group<a,b,c|a^12=b^6=1,c^2=b^3,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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