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## G = C3×C12.29D6order 432 = 24·33

### Direct product of C3 and C12.29D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C12.29D6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C32×C3⋊C8 — C3×C12.29D6
 Lower central C32 — C3×C12.29D6
 Upper central C1 — C12

Generators and relations for C3×C12.29D6
G = < a,b,c,d | a3=b12=1, c6=b3, d2=b6, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b6c5 >

Subgroups: 384 in 126 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, C2×C4, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C4×S3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, S3×C8, C2×C24, C3×C3⋊S3, C32×C6, C3×C3⋊C8, C3×C3⋊C8, C3×C24, S3×C12, C4×C3⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C12.29D6, S3×C24, C32×C3⋊C8, C12×C3⋊S3, C3×C12.29D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C12, D6, C2×C6, C2×C8, C3×S3, C24, C4×S3, C2×C12, S32, S3×C6, S3×C8, C2×C24, C6.D6, S3×C12, C3×S32, C12.29D6, S3×C24, C3×C6.D6, C3×C12.29D6

Smallest permutation representation of C3×C12.29D6
On 48 points
Generators in S48
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 48)
(1 3 5 7 9 11 13 15 17 19 21 23)(2 12 22 8 18 4 14 24 10 20 6 16)(25 35 45 31 41 27 37 47 33 43 29 39)(26 28 30 32 34 36 38 40 42 44 46 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 26 13 38)(2 43 14 31)(3 36 15 48)(4 29 16 41)(5 46 17 34)(6 39 18 27)(7 32 19 44)(8 25 20 37)(9 42 21 30)(10 35 22 47)(11 28 23 40)(12 45 24 33)

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

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

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

108 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 4D 6A 6B 6C ··· 6H 6I 6J 6K 6L 6M 6N 6O 8A ··· 8H 12A 12B 12C 12D 12E ··· 12P 12Q ··· 12V 12W 12X 12Y 12Z 24A ··· 24P 24Q ··· 24AN order 1 2 2 2 3 3 3 ··· 3 3 3 3 4 4 4 4 6 6 6 ··· 6 6 6 6 6 6 6 6 8 ··· 8 12 12 12 12 12 ··· 12 12 ··· 12 12 12 12 12 24 ··· 24 24 ··· 24 size 1 1 9 9 1 1 2 ··· 2 4 4 4 1 1 9 9 1 1 2 ··· 2 4 4 4 9 9 9 9 3 ··· 3 1 1 1 1 2 ··· 2 4 ··· 4 9 9 9 9 3 ··· 3 6 ··· 6

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 S3 D6 C3×S3 C4×S3 S3×C6 S3×C8 S3×C12 S3×C24 S32 C6.D6 C3×S32 C12.29D6 C3×C6.D6 C3×C12.29D6 kernel C3×C12.29D6 C32×C3⋊C8 C12×C3⋊S3 C12.29D6 C3×C3⋊Dic3 C6×C3⋊S3 C3×C3⋊C8 C4×C3⋊S3 C3×C3⋊S3 C3⋊Dic3 C2×C3⋊S3 C3⋊S3 C3×C3⋊C8 C3×C12 C3⋊C8 C3×C6 C12 C32 C6 C3 C12 C6 C4 C3 C2 C1 # reps 1 2 1 2 2 2 4 2 8 4 4 16 2 2 4 4 4 8 8 16 1 1 2 2 2 4

Matrix representation of C3×C12.29D6 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 27 46 0 0 27 0
,
 72 1 0 0 72 0 0 0 0 0 0 22 0 0 22 0
,
 72 0 0 0 72 1 0 0 0 0 0 46 0 0 46 0
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,27,27,0,0,46,0],[72,72,0,0,1,0,0,0,0,0,0,22,0,0,22,0],[72,72,0,0,0,1,0,0,0,0,0,46,0,0,46,0] >;

C3×C12.29D6 in GAP, Magma, Sage, TeX

C_3\times C_{12}._{29}D_6
% in TeX

G:=Group("C3xC12.29D6");
// GroupNames label

G:=SmallGroup(432,415);
// by ID

G=gap.SmallGroup(432,415);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,92,80,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=1,c^6=b^3,d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^6*c^5>;
// generators/relations

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