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## G = C6×Dic6order 144 = 24·32

### Direct product of C6 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C6×Dic6
 Chief series C1 — C3 — C6 — C3×C6 — C3×Dic3 — C6×Dic3 — C6×Dic6
 Lower central C3 — C6 — C6×Dic6
 Upper central C1 — C2×C6 — C2×C12

Generators and relations for C6×Dic6
G = < a,b,c | a6=b12=1, c2=b6, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 136 in 84 conjugacy classes, 54 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], Q8 [×4], C32, Dic3 [×4], C12 [×4], C12 [×6], C2×C6 [×2], C2×C6, C2×Q8, C3×C6, C3×C6 [×2], Dic6 [×4], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×3], C3×Q8 [×4], C3×Dic3 [×4], C3×C12 [×2], C62, C2×Dic6, C6×Q8, C3×Dic6 [×4], C6×Dic3 [×2], C6×C12, C6×Dic6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], Q8 [×2], C23, D6 [×3], C2×C6 [×7], C2×Q8, C3×S3, Dic6 [×2], C3×Q8 [×2], C22×S3, C22×C6, S3×C6 [×3], C2×Dic6, C6×Q8, C3×Dic6 [×2], S3×C2×C6, C6×Dic6

Smallest permutation representation of C6×Dic6
On 48 points
Generators in S48
(1 33 9 29 5 25)(2 34 10 30 6 26)(3 35 11 31 7 27)(4 36 12 32 8 28)(13 39 17 43 21 47)(14 40 18 44 22 48)(15 41 19 45 23 37)(16 42 20 46 24 38)
(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 19 7 13)(2 18 8 24)(3 17 9 23)(4 16 10 22)(5 15 11 21)(6 14 12 20)(25 41 31 47)(26 40 32 46)(27 39 33 45)(28 38 34 44)(29 37 35 43)(30 48 36 42)

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

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

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

54 conjugacy classes

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

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 Q8 D6 D6 C3×S3 Dic6 C3×Q8 S3×C6 S3×C6 C3×Dic6 kernel C6×Dic6 C3×Dic6 C6×Dic3 C6×C12 C2×Dic6 Dic6 C2×Dic3 C2×C12 C2×C12 C3×C6 C12 C2×C6 C2×C4 C6 C6 C4 C22 C2 # reps 1 4 2 1 2 8 4 2 1 2 2 1 2 4 4 4 2 8

Matrix representation of C6×Dic6 in GL4(𝔽13) generated by

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

C6×Dic6 in GAP, Magma, Sage, TeX

C_6\times {\rm Dic}_6
% in TeX

G:=Group("C6xDic6");
// GroupNames label

G:=SmallGroup(144,158);
// by ID

G=gap.SmallGroup(144,158);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-3,144,506,122,3461]);
// Polycyclic

G:=Group<a,b,c|a^6=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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