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G = C12×D15order 360 = 23·32·5

Direct product of C12 and D15

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C12×D15, C602C6, C605S3, D30.2C6, C30.52D6, C6.20D30, Dic155C6, C53(S3×C12), C202(C3×S3), (C3×C60)⋊4C2, C32(D5×C12), (C3×C12)⋊3D5, C122(C3×D5), C6.9(C6×D5), C1514(C4×S3), C327(C4×D5), C157(C2×C12), C10.9(S3×C6), C30.9(C2×C6), C2.1(C6×D15), (C6×D15).4C2, (C3×C6).28D10, (C3×Dic15)⋊11C2, (C3×C30).38C22, (C3×C15)⋊27(C2×C4), SmallGroup(360,101)

Series: Derived Chief Lower central Upper central

C1C15 — C12×D15
C1C5C15C30C3×C30C6×D15 — C12×D15
C15 — C12×D15
C1C12

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, C3, C3, C4, C4, C22, C5, S3, C6, C6, C2×C4, C32, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C15, C3×S3, C3×C6, Dic5, C20, D10, C4×S3, C2×C12, C3×D5, D15, C30, C30, C3×Dic3, C3×C12, S3×C6, C4×D5, C3×C15, C3×Dic5, Dic15, C60, C60, C6×D5, D30, S3×C12, C3×D15, C3×C30, D5×C12, C4×D15, C3×Dic15, C3×C60, C6×D15, C12×D15
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D5, C12, 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 59 26 32 6 49 16 37 11 54 21 42)(2 60 27 33 7 50 17 38 12 55 22 43)(3 46 28 34 8 51 18 39 13 56 23 44)(4 47 29 35 9 52 19 40 14 57 24 45)(5 48 30 36 10 53 20 41 15 58 25 31)(61 114 88 92 71 109 83 102 66 119 78 97)(62 115 89 93 72 110 84 103 67 120 79 98)(63 116 90 94 73 111 85 104 68 106 80 99)(64 117 76 95 74 112 86 105 69 107 81 100)(65 118 77 96 75 113 87 91 70 108 82 101)
(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 81)(2 80)(3 79)(4 78)(5 77)(6 76)(7 90)(8 89)(9 88)(10 87)(11 86)(12 85)(13 84)(14 83)(15 82)(16 74)(17 73)(18 72)(19 71)(20 70)(21 69)(22 68)(23 67)(24 66)(25 65)(26 64)(27 63)(28 62)(29 61)(30 75)(31 118)(32 117)(33 116)(34 115)(35 114)(36 113)(37 112)(38 111)(39 110)(40 109)(41 108)(42 107)(43 106)(44 120)(45 119)(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)| (1,59,26,32,6,49,16,37,11,54,21,42)(2,60,27,33,7,50,17,38,12,55,22,43)(3,46,28,34,8,51,18,39,13,56,23,44)(4,47,29,35,9,52,19,40,14,57,24,45)(5,48,30,36,10,53,20,41,15,58,25,31)(61,114,88,92,71,109,83,102,66,119,78,97)(62,115,89,93,72,110,84,103,67,120,79,98)(63,116,90,94,73,111,85,104,68,106,80,99)(64,117,76,95,74,112,86,105,69,107,81,100)(65,118,77,96,75,113,87,91,70,108,82,101), (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), (1,81)(2,80)(3,79)(4,78)(5,77)(6,76)(7,90)(8,89)(9,88)(10,87)(11,86)(12,85)(13,84)(14,83)(15,82)(16,74)(17,73)(18,72)(19,71)(20,70)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(29,61)(30,75)(31,118)(32,117)(33,116)(34,115)(35,114)(36,113)(37,112)(38,111)(39,110)(40,109)(41,108)(42,107)(43,106)(44,120)(45,119)(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( (1,59,26,32,6,49,16,37,11,54,21,42)(2,60,27,33,7,50,17,38,12,55,22,43)(3,46,28,34,8,51,18,39,13,56,23,44)(4,47,29,35,9,52,19,40,14,57,24,45)(5,48,30,36,10,53,20,41,15,58,25,31)(61,114,88,92,71,109,83,102,66,119,78,97)(62,115,89,93,72,110,84,103,67,120,79,98)(63,116,90,94,73,111,85,104,68,106,80,99)(64,117,76,95,74,112,86,105,69,107,81,100)(65,118,77,96,75,113,87,91,70,108,82,101), (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), (1,81)(2,80)(3,79)(4,78)(5,77)(6,76)(7,90)(8,89)(9,88)(10,87)(11,86)(12,85)(13,84)(14,83)(15,82)(16,74)(17,73)(18,72)(19,71)(20,70)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(29,61)(30,75)(31,118)(32,117)(33,116)(34,115)(35,114)(36,113)(37,112)(38,111)(39,110)(40,109)(41,108)(42,107)(43,106)(44,120)(45,119)(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([[(1,59,26,32,6,49,16,37,11,54,21,42),(2,60,27,33,7,50,17,38,12,55,22,43),(3,46,28,34,8,51,18,39,13,56,23,44),(4,47,29,35,9,52,19,40,14,57,24,45),(5,48,30,36,10,53,20,41,15,58,25,31),(61,114,88,92,71,109,83,102,66,119,78,97),(62,115,89,93,72,110,84,103,67,120,79,98),(63,116,90,94,73,111,85,104,68,106,80,99),(64,117,76,95,74,112,86,105,69,107,81,100),(65,118,77,96,75,113,87,91,70,108,82,101)], [(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)], [(1,81),(2,80),(3,79),(4,78),(5,77),(6,76),(7,90),(8,89),(9,88),(10,87),(11,86),(12,85),(13,84),(14,83),(15,82),(16,74),(17,73),(18,72),(19,71),(20,70),(21,69),(22,68),(23,67),(24,66),(25,65),(26,64),(27,63),(28,62),(29,61),(30,75),(31,118),(32,117),(33,116),(34,115),(35,114),(36,113),(37,112),(38,111),(39,110),(40,109),(41,108),(42,107),(43,106),(44,120),(45,119),(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 2A2B2C3A3B3C3D3E4A4B4C4D5A5B6A6B6C6D6E6F6G6H6I10A10B12A12B12C12D12E···12J12K12L12M12N15A···15P20A20B20C20D30A···30P60A···60AF
order12223333344445566666666610101212121212···121212121215···152020202030···3060···60
size111515112221115152211222151515152211112···2151515152···222222···22···2

108 irreducible representations

dim1111111111222222222222222222
type++++++++++
imageC1C2C2C2C3C4C6C6C6C12S3D5D6C3×S3D10C4×S3C3×D5D15S3×C6C4×D5C6×D5D30S3×C12C3×D15D5×C12C4×D15C6×D15C12×D15
kernelC12×D15C3×Dic15C3×C60C6×D15C4×D15C3×D15Dic15C60D30D15C60C3×C12C30C20C3×C6C15C12C12C10C32C6C6C5C4C3C3C2C1
# reps11112422281212224424444888816

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

40000
04000
00110
00011
,
47000
01300
0090
00634
,
04800
14000
00531
001356
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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