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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 [×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 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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