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G = C15xD12order 360 = 23·32·5

Direct product of C15 and D12

direct product, metacyclic, supersoluble, monomial

Aliases: C15xD12, C60:4C6, C60:7S3, C12:1C30, D6:1C30, C30.69D6, C4:(S3xC15), C20:3(C3xS3), C12:3(C5xS3), (C3xC60):7C2, C3:1(D4xC15), C15:6(C3xD4), (C3xC15):20D4, (S3xC30):9C2, (S3xC6):3C10, (S3xC10):4C6, C32:4(C5xD4), (C3xC12):2C10, C2.4(S3xC30), C6.3(C2xC30), C6.19(S3xC10), C10.15(S3xC6), C30.26(C2xC6), (C3xC30).49C22, (C3xC6).8(C2xC10), SmallGroup(360,97)

Series: Derived Chief Lower central Upper central

C1C6 — C15xD12
C1C3C6C30C3xC30S3xC30 — C15xD12
C3C6 — C15xD12
C1C30C60

Generators and relations for C15xD12
 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, C2xC6, C15, C15, C3xS3, C3xC6, C20, C2xC10, D12, C3xD4, C5xS3, C30, C30, C3xC12, S3xC6, C5xD4, C3xC15, C60, C60, S3xC10, C2xC30, C3xD12, S3xC15, C3xC30, C5xD12, D4xC15, C3xC60, S3xC30, C15xD12
Quotients: C1, C2, C3, C22, C5, S3, C6, D4, C10, D6, C2xC6, C15, C3xS3, C2xC10, D12, C3xD4, C5xS3, C30, S3xC6, C5xD4, S3xC10, C2xC30, C3xD12, S3xC15, C5xD12, D4xC15, S3xC30, C15xD12

Smallest permutation representation of C15xD12
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 2A2B2C3A3B3C3D3E 4 5A5B5C5D6A6B6C6D6E6F6G6H6I10A10B10C10D10E···10L12A···12H15A···15H15I···15T20A20B20C20D30A···30H30I···30T30U···30AJ60A···60AF
order122233333455556666666661010101010···1012···1215···1515···152020202030···3030···3030···3060···60
size1166112222111111222666611116···62···21···12···222221···12···26···62···2

135 irreducible representations

dim1111111111112222222222222222
type+++++++
imageC1C2C2C3C5C6C6C10C10C15C30C30S3D4D6C3xS3D12C3xD4C5xS3S3xC6C5xD4S3xC10C3xD12S3xC15C5xD12D4xC15S3xC30C15xD12
kernelC15xD12C3xC60S3xC30C5xD12C3xD12C60S3xC10C3xC12S3xC6D12C12D6C60C3xC15C30C20C15C15C12C10C32C6C5C4C3C3C2C1
# reps112242448881611122242444888816

Matrix representation of C15xD12 in GL2(F61) generated by

160
016
,
210
032
,
032
210
G:=sub<GL(2,GF(61))| [16,0,0,16],[21,0,0,32],[0,21,32,0] >;

C15xD12 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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