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## G = C10×D12order 240 = 24·3·5

### Direct product of C10 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C10×D12
 Chief series C1 — C3 — C6 — C30 — S3×C10 — S3×C2×C10 — C10×D12
 Lower central C3 — C6 — C10×D12
 Upper central C1 — C2×C10 — C2×C20

Generators and relations for C10×D12
G = < a,b,c | a10=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 248 in 108 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C10, C10, C10, C12, D6, D6, C2×C6, C15, C2×D4, C20, C2×C10, C2×C10, D12, C2×C12, C22×S3, C5×S3, C30, C30, C2×C20, C5×D4, C22×C10, C2×D12, C60, S3×C10, S3×C10, C2×C30, D4×C10, C5×D12, C2×C60, S3×C2×C10, C10×D12
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, C2×D4, C2×C10, D12, C22×S3, C5×S3, C5×D4, C22×C10, C2×D12, S3×C10, D4×C10, C5×D12, S3×C2×C10, C10×D12

Smallest permutation representation of C10×D12
On 120 points
Generators in S120
(1 71 30 22 96 74 108 39 50 113)(2 72 31 23 85 75 97 40 51 114)(3 61 32 24 86 76 98 41 52 115)(4 62 33 13 87 77 99 42 53 116)(5 63 34 14 88 78 100 43 54 117)(6 64 35 15 89 79 101 44 55 118)(7 65 36 16 90 80 102 45 56 119)(8 66 25 17 91 81 103 46 57 120)(9 67 26 18 92 82 104 47 58 109)(10 68 27 19 93 83 105 48 59 110)(11 69 28 20 94 84 106 37 60 111)(12 70 29 21 95 73 107 38 49 112)
(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)(25 34)(26 33)(27 32)(28 31)(29 30)(35 36)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)(49 50)(51 60)(52 59)(53 58)(54 57)(55 56)(61 68)(62 67)(63 66)(64 65)(69 72)(70 71)(73 74)(75 84)(76 83)(77 82)(78 81)(79 80)(85 94)(86 93)(87 92)(88 91)(89 90)(95 96)(97 106)(98 105)(99 104)(100 103)(101 102)(107 108)(109 116)(110 115)(111 114)(112 113)(117 120)(118 119)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 5C 5D 6A 6B 6C 10A ··· 10L 10M ··· 10AB 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 30A ··· 30L 60A ··· 60P order 1 2 2 2 2 2 2 2 3 4 4 5 5 5 5 6 6 6 10 ··· 10 10 ··· 10 12 12 12 12 15 15 15 15 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 6 6 6 6 2 2 2 1 1 1 1 2 2 2 1 ··· 1 6 ··· 6 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 S3 D4 D6 D6 D12 C5×S3 C5×D4 S3×C10 S3×C10 C5×D12 kernel C10×D12 C5×D12 C2×C60 S3×C2×C10 C2×D12 D12 C2×C12 C22×S3 C2×C20 C30 C20 C2×C10 C10 C2×C4 C6 C4 C22 C2 # reps 1 4 1 2 4 16 4 8 1 2 2 1 4 4 8 8 4 16

Matrix representation of C10×D12 in GL4(𝔽61) generated by

 27 0 0 0 0 27 0 0 0 0 9 0 0 0 0 9
,
 1 1 0 0 60 0 0 0 0 0 23 38 0 0 23 46
,
 1 1 0 0 0 60 0 0 0 0 38 23 0 0 46 23
G:=sub<GL(4,GF(61))| [27,0,0,0,0,27,0,0,0,0,9,0,0,0,0,9],[1,60,0,0,1,0,0,0,0,0,23,23,0,0,38,46],[1,0,0,0,1,60,0,0,0,0,38,46,0,0,23,23] >;

C10×D12 in GAP, Magma, Sage, TeX

C_{10}\times D_{12}
% in TeX

G:=Group("C10xD12");
// GroupNames label

G:=SmallGroup(240,167);
// by ID

G=gap.SmallGroup(240,167);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-3,794,194,5765]);
// Polycyclic

G:=Group<a,b,c|a^10=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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