Copied to
clipboard

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 [×2], C2 [×12], C3, C4 [×2], C4 [×2], C22, C22 [×38], C5, S3 [×8], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×5], D4 [×16], C23 [×21], D5 [×4], D5 [×4], C10, C10 [×2], C10 [×4], C12 [×2], C12 [×2], D6 [×4], D6 [×28], C2×C6, C2×C6 [×6], C15, C22×C4, C2×D4 [×12], C24 [×2], Dic5 [×2], C20 [×2], D10 [×6], D10 [×24], C2×C10, C2×C10 [×8], D12 [×4], D12 [×12], C2×C12, C2×C12 [×5], C22×S3 [×2], C22×S3 [×18], C22×C6, C5×S3 [×4], C3×D5 [×4], D15 [×4], C30, C30 [×2], C22×D4, C4×D5 [×4], D20 [×4], C2×Dic5, C5⋊D4 [×8], C2×C20, C5×D4 [×4], C22×D5, C22×D5 [×18], C22×C10 [×2], C2×D12, C2×D12 [×11], C22×C12, S3×C23 [×2], C3×Dic5 [×2], C60 [×2], S3×D5 [×16], C6×D5 [×6], S3×C10 [×4], S3×C10 [×4], D30 [×4], D30 [×4], C2×C30, C2×C4×D5, C2×D20, D4×D5 [×8], C2×C5⋊D4 [×2], D4×C10, C23×D5 [×2], C22×D12, C5⋊D12 [×8], D5×C12 [×4], C6×Dic5, C5×D12 [×4], D60 [×4], C2×C60, C2×S3×D5 [×8], C2×S3×D5 [×8], D5×C2×C6, S3×C2×C10 [×2], C22×D15 [×2], C2×D4×D5, D5×D12 [×8], C2×C5⋊D12 [×2], D5×C2×C12, C10×D12, C2×D60, C22×S3×D5 [×2], C2×D5×D12
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D5, D6 [×7], C2×D4 [×6], C24, D10 [×7], D12 [×4], C22×S3 [×7], C22×D4, C22×D5 [×7], C2×D12 [×6], S3×C23, S3×D5, D4×D5 [×2], C23×D5, C22×D12, C2×S3×D5 [×3], C2×D4×D5, D5×D12 [×2], C22×S3×D5, C2×D5×D12

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

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

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

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

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

׿
×
𝔽