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## G = C12×D15order 360 = 23·32·5

### Direct product of C12 and D15

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C12×D15
 Chief series C1 — C5 — C15 — C30 — C3×C30 — C6×D15 — C12×D15
 Lower central C15 — C12×D15
 Upper central C1 — C12

Generators and relations for C12×D15
G = < a,b,c | a12=b15=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 276 in 70 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2 [×2], C3 [×2], C3, C4, C4, C22, C5, S3 [×2], C6 [×2], C6 [×3], C2×C4, C32, D5 [×2], C10, Dic3, C12 [×2], C12 [×2], D6, C2×C6, C15 [×2], C15, C3×S3 [×2], C3×C6, Dic5, C20, D10, C4×S3, C2×C12, C3×D5 [×2], D15 [×2], C30 [×2], C30, C3×Dic3, C3×C12, S3×C6, C4×D5, C3×C15, C3×Dic5, Dic15, C60 [×2], C60, C6×D5, D30, S3×C12, C3×D15 [×2], C3×C30, D5×C12, C4×D15, C3×Dic15, C3×C60, C6×D15, C12×D15
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D5, C12 [×2], D6, C2×C6, C3×S3, D10, C4×S3, C2×C12, C3×D5, D15, S3×C6, C4×D5, C6×D5, D30, S3×C12, C3×D15, D5×C12, C4×D15, C6×D15, C12×D15

Smallest permutation representation of C12×D15
On 120 points
Generators in S120
(1 55 19 32 6 60 24 37 11 50 29 42)(2 56 20 33 7 46 25 38 12 51 30 43)(3 57 21 34 8 47 26 39 13 52 16 44)(4 58 22 35 9 48 27 40 14 53 17 45)(5 59 23 36 10 49 28 41 15 54 18 31)(61 120 79 95 71 115 89 105 66 110 84 100)(62 106 80 96 72 116 90 91 67 111 85 101)(63 107 81 97 73 117 76 92 68 112 86 102)(64 108 82 98 74 118 77 93 69 113 87 103)(65 109 83 99 75 119 78 94 70 114 88 104)
(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 97 98 99 100 101 102 103 104 105)(106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)
(1 88)(2 87)(3 86)(4 85)(5 84)(6 83)(7 82)(8 81)(9 80)(10 79)(11 78)(12 77)(13 76)(14 90)(15 89)(16 68)(17 67)(18 66)(19 65)(20 64)(21 63)(22 62)(23 61)(24 75)(25 74)(26 73)(27 72)(28 71)(29 70)(30 69)(31 110)(32 109)(33 108)(34 107)(35 106)(36 120)(37 119)(38 118)(39 117)(40 116)(41 115)(42 114)(43 113)(44 112)(45 111)(46 98)(47 97)(48 96)(49 95)(50 94)(51 93)(52 92)(53 91)(54 105)(55 104)(56 103)(57 102)(58 101)(59 100)(60 99)

G:=sub<Sym(120)| (1,55,19,32,6,60,24,37,11,50,29,42)(2,56,20,33,7,46,25,38,12,51,30,43)(3,57,21,34,8,47,26,39,13,52,16,44)(4,58,22,35,9,48,27,40,14,53,17,45)(5,59,23,36,10,49,28,41,15,54,18,31)(61,120,79,95,71,115,89,105,66,110,84,100)(62,106,80,96,72,116,90,91,67,111,85,101)(63,107,81,97,73,117,76,92,68,112,86,102)(64,108,82,98,74,118,77,93,69,113,87,103)(65,109,83,99,75,119,78,94,70,114,88,104), (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)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,88)(2,87)(3,86)(4,85)(5,84)(6,83)(7,82)(8,81)(9,80)(10,79)(11,78)(12,77)(13,76)(14,90)(15,89)(16,68)(17,67)(18,66)(19,65)(20,64)(21,63)(22,62)(23,61)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,110)(32,109)(33,108)(34,107)(35,106)(36,120)(37,119)(38,118)(39,117)(40,116)(41,115)(42,114)(43,113)(44,112)(45,111)(46,98)(47,97)(48,96)(49,95)(50,94)(51,93)(52,92)(53,91)(54,105)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99)>;

G:=Group( (1,55,19,32,6,60,24,37,11,50,29,42)(2,56,20,33,7,46,25,38,12,51,30,43)(3,57,21,34,8,47,26,39,13,52,16,44)(4,58,22,35,9,48,27,40,14,53,17,45)(5,59,23,36,10,49,28,41,15,54,18,31)(61,120,79,95,71,115,89,105,66,110,84,100)(62,106,80,96,72,116,90,91,67,111,85,101)(63,107,81,97,73,117,76,92,68,112,86,102)(64,108,82,98,74,118,77,93,69,113,87,103)(65,109,83,99,75,119,78,94,70,114,88,104), (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)(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105)(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120), (1,88)(2,87)(3,86)(4,85)(5,84)(6,83)(7,82)(8,81)(9,80)(10,79)(11,78)(12,77)(13,76)(14,90)(15,89)(16,68)(17,67)(18,66)(19,65)(20,64)(21,63)(22,62)(23,61)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,110)(32,109)(33,108)(34,107)(35,106)(36,120)(37,119)(38,118)(39,117)(40,116)(41,115)(42,114)(43,113)(44,112)(45,111)(46,98)(47,97)(48,96)(49,95)(50,94)(51,93)(52,92)(53,91)(54,105)(55,104)(56,103)(57,102)(58,101)(59,100)(60,99) );

G=PermutationGroup([(1,55,19,32,6,60,24,37,11,50,29,42),(2,56,20,33,7,46,25,38,12,51,30,43),(3,57,21,34,8,47,26,39,13,52,16,44),(4,58,22,35,9,48,27,40,14,53,17,45),(5,59,23,36,10,49,28,41,15,54,18,31),(61,120,79,95,71,115,89,105,66,110,84,100),(62,106,80,96,72,116,90,91,67,111,85,101),(63,107,81,97,73,117,76,92,68,112,86,102),(64,108,82,98,74,118,77,93,69,113,87,103),(65,109,83,99,75,119,78,94,70,114,88,104)], [(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),(91,92,93,94,95,96,97,98,99,100,101,102,103,104,105),(106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)], [(1,88),(2,87),(3,86),(4,85),(5,84),(6,83),(7,82),(8,81),(9,80),(10,79),(11,78),(12,77),(13,76),(14,90),(15,89),(16,68),(17,67),(18,66),(19,65),(20,64),(21,63),(22,62),(23,61),(24,75),(25,74),(26,73),(27,72),(28,71),(29,70),(30,69),(31,110),(32,109),(33,108),(34,107),(35,106),(36,120),(37,119),(38,118),(39,117),(40,116),(41,115),(42,114),(43,113),(44,112),(45,111),(46,98),(47,97),(48,96),(49,95),(50,94),(51,93),(52,92),(53,91),(54,105),(55,104),(56,103),(57,102),(58,101),(59,100),(60,99)])

108 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 6F 6G 6H 6I 10A 10B 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 15A ··· 15P 20A 20B 20C 20D 30A ··· 30P 60A ··· 60AF order 1 2 2 2 3 3 3 3 3 4 4 4 4 5 5 6 6 6 6 6 6 6 6 6 10 10 12 12 12 12 12 ··· 12 12 12 12 12 15 ··· 15 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 15 15 1 1 2 2 2 1 1 15 15 2 2 1 1 2 2 2 15 15 15 15 2 2 1 1 1 1 2 ··· 2 15 15 15 15 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2

108 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 S3 D5 D6 C3×S3 D10 C4×S3 C3×D5 D15 S3×C6 C4×D5 C6×D5 D30 S3×C12 C3×D15 D5×C12 C4×D15 C6×D15 C12×D15 kernel C12×D15 C3×Dic15 C3×C60 C6×D15 C4×D15 C3×D15 Dic15 C60 D30 D15 C60 C3×C12 C30 C20 C3×C6 C15 C12 C12 C10 C32 C6 C6 C5 C4 C3 C3 C2 C1 # reps 1 1 1 1 2 4 2 2 2 8 1 2 1 2 2 2 4 4 2 4 4 4 4 8 8 8 8 16

Matrix representation of C12×D15 in GL4(𝔽61) generated by

 40 0 0 0 0 40 0 0 0 0 11 0 0 0 0 11
,
 47 0 0 0 0 13 0 0 0 0 9 0 0 0 6 34
,
 0 48 0 0 14 0 0 0 0 0 5 31 0 0 13 56
G:=sub<GL(4,GF(61))| [40,0,0,0,0,40,0,0,0,0,11,0,0,0,0,11],[47,0,0,0,0,13,0,0,0,0,9,6,0,0,0,34],[0,14,0,0,48,0,0,0,0,0,5,13,0,0,31,56] >;

C12×D15 in GAP, Magma, Sage, TeX

C_{12}\times D_{15}
% in TeX

G:=Group("C12xD15");
// GroupNames label

G:=SmallGroup(360,101);
// by ID

G=gap.SmallGroup(360,101);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-5,79,1444,10373]);
// Polycyclic

G:=Group<a,b,c|a^12=b^15=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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