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## G = C3×Dic12order 144 = 24·32

### Direct product of C3 and Dic12

Aliases: C3×Dic12, C24.1C6, C24.5S3, C324Q16, C12.62D6, C6.21D12, Dic6.1C6, C8.(C3×S3), C31(C3×Q16), C6.3(C3×D4), C4.10(S3×C6), (C3×C24).2C2, (C3×C6).19D4, C2.5(C3×D12), C12.10(C2×C6), (C3×Dic6).4C2, (C3×C12).39C22, SmallGroup(144,73)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C3×Dic12
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C3×Dic6 — C3×Dic12
 Lower central C3 — C6 — C12 — C3×Dic12
 Upper central C1 — C6 — C12 — C24

Generators and relations for C3×Dic12
G = < a,b,c | a3=b24=1, c2=b12, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C3×Dic12
On 48 points
Generators in S48
(1 9 17)(2 10 18)(3 11 19)(4 12 20)(5 13 21)(6 14 22)(7 15 23)(8 16 24)(25 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)
(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 37 13 25)(2 36 14 48)(3 35 15 47)(4 34 16 46)(5 33 17 45)(6 32 18 44)(7 31 19 43)(8 30 20 42)(9 29 21 41)(10 28 22 40)(11 27 23 39)(12 26 24 38)

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

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

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

C3×Dic12 is a maximal subgroup of
D24.S3  C24.49D6  C322Q32  C323Q32  Dic12⋊S3  C24.23D6  D6.3D12  D245S3  D12.4D6  C3×S3×Q16  He34Q16  C72.C6  He35Q16
C3×Dic12 is a maximal quotient of
He34Q16  C72.C6

45 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 6D 6E 8A 8B 12A ··· 12H 12I 12J 12K 12L 24A ··· 24P order 1 2 3 3 3 3 3 4 4 4 6 6 6 6 6 8 8 12 ··· 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 2 2 12 12 1 1 2 2 2 2 2 2 ··· 2 12 12 12 12 2 ··· 2

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - image C1 C2 C2 C3 C6 C6 S3 D4 D6 Q16 C3×S3 D12 C3×D4 S3×C6 Dic12 C3×Q16 C3×D12 C3×Dic12 kernel C3×Dic12 C3×C24 C3×Dic6 Dic12 C24 Dic6 C24 C3×C6 C12 C32 C8 C6 C6 C4 C3 C3 C2 C1 # reps 1 1 2 2 2 4 1 1 1 2 2 2 2 2 4 4 4 8

Matrix representation of C3×Dic12 in GL2(𝔽73) generated by

 64 0 0 64
,
 43 0 0 17
,
 0 1 72 0
G:=sub<GL(2,GF(73))| [64,0,0,64],[43,0,0,17],[0,72,1,0] >;

C3×Dic12 in GAP, Magma, Sage, TeX

C_3\times {\rm Dic}_{12}
% in TeX

G:=Group("C3xDic12");
// GroupNames label

G:=SmallGroup(144,73);
// by ID

G=gap.SmallGroup(144,73);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-3,144,169,223,867,69,3461]);
// Polycyclic

G:=Group<a,b,c|a^3=b^24=1,c^2=b^12,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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