Copied to
clipboard

## 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, 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 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

׿
×
𝔽