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## G = C2×D5×D12order 480 = 25·3·5

### Direct product of C2, D5 and D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×D5×D12
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — C22×S3×D5 — C2×D5×D12
 Lower central C15 — C30 — C2×D5×D12
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×D5×D12
G = < a,b,c,d,e | a2=b5=c2=d12=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 2972 in 472 conjugacy classes, 132 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, D5, D5, C10, C10, C10, C12, C12, D6, D6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, D12, D12, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, C3×D5, D15, C30, C30, C22×D4, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C2×D12, C2×D12, C22×C12, S3×C23, C3×Dic5, C60, S3×D5, C6×D5, S3×C10, S3×C10, D30, D30, C2×C30, C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, C22×D12, C5⋊D12, D5×C12, C6×Dic5, C5×D12, D60, C2×C60, C2×S3×D5, C2×S3×D5, D5×C2×C6, S3×C2×C10, C22×D15, C2×D4×D5, D5×D12, C2×C5⋊D12, D5×C2×C12, C10×D12, C2×D60, C22×S3×D5, C2×D5×D12
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C24, D10, D12, C22×S3, C22×D4, C22×D5, C2×D12, S3×C23, S3×D5, D4×D5, C23×D5, C22×D12, C2×S3×D5, C2×D4×D5, D5×D12, C22×S3×D5, C2×D5×D12

Smallest permutation representation of C2×D5×D12
On 120 points
Generators in S120
(1 105)(2 106)(3 107)(4 108)(5 97)(6 98)(7 99)(8 100)(9 101)(10 102)(11 103)(12 104)(13 75)(14 76)(15 77)(16 78)(17 79)(18 80)(19 81)(20 82)(21 83)(22 84)(23 73)(24 74)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 61)(33 62)(34 63)(35 64)(36 65)(37 120)(38 109)(39 110)(40 111)(41 112)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(57 85)(58 86)(59 87)(60 88)
(1 71 19 53 37)(2 72 20 54 38)(3 61 21 55 39)(4 62 22 56 40)(5 63 23 57 41)(6 64 24 58 42)(7 65 13 59 43)(8 66 14 60 44)(9 67 15 49 45)(10 68 16 50 46)(11 69 17 51 47)(12 70 18 52 48)(25 76 88 115 100)(26 77 89 116 101)(27 78 90 117 102)(28 79 91 118 103)(29 80 92 119 104)(30 81 93 120 105)(31 82 94 109 106)(32 83 95 110 107)(33 84 96 111 108)(34 73 85 112 97)(35 74 86 113 98)(36 75 87 114 99)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 94)(26 95)(27 96)(28 85)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(97 118)(98 119)(99 120)(100 109)(101 110)(102 111)(103 112)(104 113)(105 114)(106 115)(107 116)(108 117)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 15)(16 24)(17 23)(18 22)(19 21)(26 36)(27 35)(28 34)(29 33)(30 32)(37 39)(40 48)(41 47)(42 46)(43 45)(49 59)(50 58)(51 57)(52 56)(53 55)(61 71)(62 70)(63 69)(64 68)(65 67)(73 79)(74 78)(75 77)(80 84)(81 83)(85 91)(86 90)(87 89)(92 96)(93 95)(97 103)(98 102)(99 101)(104 108)(105 107)(110 120)(111 119)(112 118)(113 117)(114 116)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 20A 20B 20C 20D 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 10 ··· 10 12 12 12 12 12 12 12 12 15 15 20 20 20 20 30 ··· 30 60 ··· 60 size 1 1 1 1 5 5 5 5 6 6 6 6 30 30 30 30 2 2 2 10 10 2 2 2 2 2 10 10 10 10 2 ··· 2 12 ··· 12 2 2 2 2 10 10 10 10 4 4 4 4 4 4 4 ··· 4 4 ··· 4

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D5 D6 D6 D6 D6 D10 D10 D10 D12 S3×D5 D4×D5 C2×S3×D5 C2×S3×D5 D5×D12 kernel C2×D5×D12 D5×D12 C2×C5⋊D12 D5×C2×C12 C10×D12 C2×D60 C22×S3×D5 C2×C4×D5 C6×D5 C2×D12 C4×D5 C2×Dic5 C2×C20 C22×D5 D12 C2×C12 C22×S3 D10 C2×C4 C6 C4 C22 C2 # reps 1 8 2 1 1 1 2 1 4 2 4 1 1 1 8 2 4 8 2 4 4 2 8

Matrix representation of C2×D5×D12 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 60 17
,
 60 0 0 0 0 60 0 0 0 0 0 1 0 0 1 0
,
 15 23 0 0 38 38 0 0 0 0 60 0 0 0 0 60
,
 60 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,1,0,0,0,0,0,60,0,0,1,17],[60,0,0,0,0,60,0,0,0,0,0,1,0,0,1,0],[15,38,0,0,23,38,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,1,1,0,0,0,0,1,0,0,0,0,1] >;

C2×D5×D12 in GAP, Magma, Sage, TeX

C_2\times D_5\times D_{12}
% in TeX

G:=Group("C2xD5xD12");
// GroupNames label

G:=SmallGroup(480,1087);
// by ID

G=gap.SmallGroup(480,1087);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,346,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^12=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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