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## G = C12×D12order 288 = 25·32

### Direct product of C12 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C12×D12
 Chief series C1 — C3 — C6 — C2×C6 — C62 — S3×C2×C6 — C6×D12 — C12×D12
 Lower central C3 — C6 — C12×D12
 Upper central C1 — C2×C12 — C4×C12

Generators and relations for C12×D12
G = < a,b,c | a12=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 466 in 203 conjugacy classes, 90 normal (42 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×4], C4 [×3], C22, C22 [×8], S3 [×4], C6 [×6], C6 [×7], C2×C4 [×3], C2×C4 [×6], D4 [×4], C23 [×2], C32, Dic3 [×2], C12 [×8], C12 [×10], D6 [×4], D6 [×4], C2×C6 [×2], C2×C6 [×9], C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, C3×S3 [×4], C3×C6 [×3], C4×S3 [×4], D12 [×4], C2×Dic3 [×2], C2×C12 [×6], C2×C12 [×9], C3×D4 [×4], C22×S3 [×2], C22×C6 [×2], C4×D4, C3×Dic3 [×2], C3×C12 [×4], C3×C12, S3×C6 [×4], S3×C6 [×4], C62, C4⋊Dic3, D6⋊C4 [×2], C4×C12 [×2], C4×C12, C3×C22⋊C4 [×2], C3×C4⋊C4, S3×C2×C4 [×2], C2×D12, C22×C12 [×2], C6×D4, S3×C12 [×4], C3×D12 [×4], C6×Dic3 [×2], C6×C12 [×3], S3×C2×C6 [×2], C4×D12, D4×C12, C3×C4⋊Dic3, C3×D6⋊C4 [×2], C122, S3×C2×C12 [×2], C6×D12, C12×D12
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], S3, C6 [×7], C2×C4 [×6], D4 [×2], C23, C12 [×4], D6 [×3], C2×C6 [×7], C22×C4, C2×D4, C4○D4, C3×S3, C4×S3 [×2], D12 [×2], C2×C12 [×6], C3×D4 [×2], C22×S3, C22×C6, C4×D4, S3×C6 [×3], S3×C2×C4, C2×D12, C4○D12, C22×C12, C6×D4, C3×C4○D4, S3×C12 [×2], C3×D12 [×2], S3×C2×C6, C4×D12, D4×C12, S3×C2×C12, C6×D12, C3×C4○D12, C12×D12

Smallest permutation representation of C12×D12
On 96 points
Generators in S96
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 82 39 25 5 74 43 29 9 78 47 33)(2 83 40 26 6 75 44 30 10 79 48 34)(3 84 41 27 7 76 45 31 11 80 37 35)(4 73 42 28 8 77 46 32 12 81 38 36)(13 90 60 66 21 86 56 62 17 94 52 70)(14 91 49 67 22 87 57 63 18 95 53 71)(15 92 50 68 23 88 58 64 19 96 54 72)(16 93 51 69 24 89 59 65 20 85 55 61)
(1 59)(2 60)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 46)(20 47)(21 48)(22 37)(23 38)(24 39)(25 69)(26 70)(27 71)(28 72)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(73 92)(74 93)(75 94)(76 95)(77 96)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)

G:=sub<Sym(96)| (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,82,39,25,5,74,43,29,9,78,47,33)(2,83,40,26,6,75,44,30,10,79,48,34)(3,84,41,27,7,76,45,31,11,80,37,35)(4,73,42,28,8,77,46,32,12,81,38,36)(13,90,60,66,21,86,56,62,17,94,52,70)(14,91,49,67,22,87,57,63,18,95,53,71)(15,92,50,68,23,88,58,64,19,96,54,72)(16,93,51,69,24,89,59,65,20,85,55,61), (1,59)(2,60)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,37)(23,38)(24,39)(25,69)(26,70)(27,71)(28,72)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(73,92)(74,93)(75,94)(76,95)(77,96)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91)>;

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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,82,39,25,5,74,43,29,9,78,47,33)(2,83,40,26,6,75,44,30,10,79,48,34)(3,84,41,27,7,76,45,31,11,80,37,35)(4,73,42,28,8,77,46,32,12,81,38,36)(13,90,60,66,21,86,56,62,17,94,52,70)(14,91,49,67,22,87,57,63,18,95,53,71)(15,92,50,68,23,88,58,64,19,96,54,72)(16,93,51,69,24,89,59,65,20,85,55,61), (1,59)(2,60)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,57)(12,58)(13,40)(14,41)(15,42)(16,43)(17,44)(18,45)(19,46)(20,47)(21,48)(22,37)(23,38)(24,39)(25,69)(26,70)(27,71)(28,72)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(73,92)(74,93)(75,94)(76,95)(77,96)(78,85)(79,86)(80,87)(81,88)(82,89)(83,90)(84,91) );

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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,82,39,25,5,74,43,29,9,78,47,33),(2,83,40,26,6,75,44,30,10,79,48,34),(3,84,41,27,7,76,45,31,11,80,37,35),(4,73,42,28,8,77,46,32,12,81,38,36),(13,90,60,66,21,86,56,62,17,94,52,70),(14,91,49,67,22,87,57,63,18,95,53,71),(15,92,50,68,23,88,58,64,19,96,54,72),(16,93,51,69,24,89,59,65,20,85,55,61)], [(1,59),(2,60),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,57),(12,58),(13,40),(14,41),(15,42),(16,43),(17,44),(18,45),(19,46),(20,47),(21,48),(22,37),(23,38),(24,39),(25,69),(26,70),(27,71),(28,72),(29,61),(30,62),(31,63),(32,64),(33,65),(34,66),(35,67),(36,68),(73,92),(74,93),(75,94),(76,95),(77,96),(78,85),(79,86),(80,87),(81,88),(82,89),(83,90),(84,91)])

108 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A ··· 6F 6G ··· 6O 6P ··· 6W 12A ··· 12H 12I ··· 12AZ 12BA ··· 12BH order 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 ··· 12 size 1 1 1 1 6 6 6 6 1 1 2 2 2 1 1 1 1 2 2 2 2 6 6 6 6 1 ··· 1 2 ··· 2 6 ··· 6 1 ··· 1 2 ··· 2 6 ··· 6

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C6 C12 S3 D4 D6 C4○D4 C3×S3 C4×S3 D12 C3×D4 S3×C6 C4○D12 C3×C4○D4 S3×C12 C3×D12 C3×C4○D12 kernel C12×D12 C3×C4⋊Dic3 C3×D6⋊C4 C122 S3×C2×C12 C6×D12 C4×D12 C3×D12 C4⋊Dic3 D6⋊C4 C4×C12 S3×C2×C4 C2×D12 D12 C4×C12 C3×C12 C2×C12 C3×C6 C42 C12 C12 C12 C2×C4 C6 C6 C4 C4 C2 # reps 1 1 2 1 2 1 2 8 2 4 2 4 2 16 1 2 3 2 2 4 4 4 6 4 4 8 8 8

Matrix representation of C12×D12 in GL4(𝔽13) generated by

 8 0 0 0 0 8 0 0 0 0 9 0 0 0 0 9
,
 8 0 0 0 0 5 0 0 0 0 6 0 0 0 0 11
,
 0 5 0 0 8 0 0 0 0 0 0 11 0 0 6 0
G:=sub<GL(4,GF(13))| [8,0,0,0,0,8,0,0,0,0,9,0,0,0,0,9],[8,0,0,0,0,5,0,0,0,0,6,0,0,0,0,11],[0,8,0,0,5,0,0,0,0,0,0,6,0,0,11,0] >;

C12×D12 in GAP, Magma, Sage, TeX

C_{12}\times D_{12}
% in TeX

G:=Group("C12xD12");
// GroupNames label

G:=SmallGroup(288,644);
// by ID

G=gap.SmallGroup(288,644);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,344,142,9414]);
// Polycyclic

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

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