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## G = C15×D12order 360 = 23·32·5

### Direct product of C15 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C15×D12
 Chief series C1 — C3 — C6 — C30 — C3×C30 — S3×C30 — C15×D12
 Lower central C3 — C6 — C15×D12
 Upper central C1 — C30 — C60

Generators and relations for C15×D12
G = < a,b,c | a15=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 148 in 70 conjugacy classes, 36 normal (28 characteristic)
C1, C2, C2, C3, C3, C4, C22, C5, S3, C6, C6, D4, C32, C10, C10, C12, C12, D6, C2×C6, C15, C15, C3×S3, C3×C6, C20, C2×C10, D12, C3×D4, C5×S3, C30, C30, C3×C12, S3×C6, C5×D4, C3×C15, C60, C60, S3×C10, C2×C30, C3×D12, S3×C15, C3×C30, C5×D12, D4×C15, C3×C60, S3×C30, C15×D12
Quotients: C1, C2, C3, C22, C5, S3, C6, D4, C10, D6, C2×C6, C15, C3×S3, C2×C10, D12, C3×D4, C5×S3, C30, S3×C6, C5×D4, S3×C10, C2×C30, C3×D12, S3×C15, C5×D12, D4×C15, S3×C30, C15×D12

Smallest permutation representation of C15×D12
On 120 points
Generators in S120
(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 35 61 83 11 45 71 78 6 40 66 88)(2 36 62 84 12 31 72 79 7 41 67 89)(3 37 63 85 13 32 73 80 8 42 68 90)(4 38 64 86 14 33 74 81 9 43 69 76)(5 39 65 87 15 34 75 82 10 44 70 77)(16 93 54 116 21 98 59 106 26 103 49 111)(17 94 55 117 22 99 60 107 27 104 50 112)(18 95 56 118 23 100 46 108 28 105 51 113)(19 96 57 119 24 101 47 109 29 91 52 114)(20 97 58 120 25 102 48 110 30 92 53 115)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 74)(17 75)(18 61)(19 62)(20 63)(21 64)(22 65)(23 66)(24 67)(25 68)(26 69)(27 70)(28 71)(29 72)(30 73)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)

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

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

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

135 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4 5A 5B 5C 5D 6A 6B 6C 6D 6E 6F 6G 6H 6I 10A 10B 10C 10D 10E ··· 10L 12A ··· 12H 15A ··· 15H 15I ··· 15T 20A 20B 20C 20D 30A ··· 30H 30I ··· 30T 30U ··· 30AJ 60A ··· 60AF order 1 2 2 2 3 3 3 3 3 4 5 5 5 5 6 6 6 6 6 6 6 6 6 10 10 10 10 10 ··· 10 12 ··· 12 15 ··· 15 15 ··· 15 20 20 20 20 30 ··· 30 30 ··· 30 30 ··· 30 60 ··· 60 size 1 1 6 6 1 1 2 2 2 2 1 1 1 1 1 1 2 2 2 6 6 6 6 1 1 1 1 6 ··· 6 2 ··· 2 1 ··· 1 2 ··· 2 2 2 2 2 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2

135 irreducible representations

 dim 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 2 2 type + + + + + + + image C1 C2 C2 C3 C5 C6 C6 C10 C10 C15 C30 C30 S3 D4 D6 C3×S3 D12 C3×D4 C5×S3 S3×C6 C5×D4 S3×C10 C3×D12 S3×C15 C5×D12 D4×C15 S3×C30 C15×D12 kernel C15×D12 C3×C60 S3×C30 C5×D12 C3×D12 C60 S3×C10 C3×C12 S3×C6 D12 C12 D6 C60 C3×C15 C30 C20 C15 C15 C12 C10 C32 C6 C5 C4 C3 C3 C2 C1 # reps 1 1 2 2 4 2 4 4 8 8 8 16 1 1 1 2 2 2 4 2 4 4 4 8 8 8 8 16

Matrix representation of C15×D12 in GL2(𝔽61) generated by

 16 0 0 16
,
 21 0 0 32
,
 0 32 21 0
G:=sub<GL(2,GF(61))| [16,0,0,16],[21,0,0,32],[0,21,32,0] >;

C15×D12 in GAP, Magma, Sage, TeX

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

G:=Group("C15xD12");
// GroupNames label

G:=SmallGroup(360,97);
// by ID

G=gap.SmallGroup(360,97);
# by ID

G:=PCGroup([6,-2,-2,-3,-5,-2,-3,745,367,8645]);
// Polycyclic

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

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