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## G = C2×S3×D20order 480 = 25·3·5

### Direct product of C2, S3 and D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×S3×D20
 Chief series C1 — C5 — C15 — C30 — C6×D5 — C2×S3×D5 — C22×S3×D5 — C2×S3×D20
 Lower central C15 — C30 — C2×S3×D20
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×S3×D20
G = < a,b,c,d,e | a2=b3=c2=d20=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: 3004 in 472 conjugacy classes, 132 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×12], C3, C4 [×2], C4 [×2], C22, C22 [×38], C5, S3 [×4], S3 [×4], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×5], D4 [×16], C23 [×21], D5 [×8], C10, C10 [×2], C10 [×4], Dic3 [×2], C12 [×2], D6 [×6], D6 [×24], C2×C6, C2×C6 [×8], C15, C22×C4, C2×D4 [×12], C24 [×2], C20 [×2], C20 [×2], D10 [×4], D10 [×28], C2×C10, C2×C10 [×6], C4×S3 [×4], D12 [×4], C2×Dic3, C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3, C22×S3 [×18], C22×C6 [×2], C5×S3 [×4], C3×D5 [×4], D15 [×4], C30, C30 [×2], C22×D4, D20 [×4], D20 [×12], C2×C20, C2×C20 [×5], C22×D5 [×2], C22×D5 [×18], C22×C10, S3×C2×C4, C2×D12, S3×D4 [×8], C2×C3⋊D4 [×2], C6×D4, S3×C23 [×2], C5×Dic3 [×2], C60 [×2], S3×D5 [×16], C6×D5 [×4], C6×D5 [×4], S3×C10 [×6], D30 [×4], D30 [×4], C2×C30, C2×D20, C2×D20 [×11], C22×C20, C23×D5 [×2], C2×S3×D4, C3⋊D20 [×8], C3×D20 [×4], S3×C20 [×4], C10×Dic3, D60 [×4], C2×C60, C2×S3×D5 [×8], C2×S3×D5 [×8], D5×C2×C6 [×2], S3×C2×C10, C22×D15 [×2], C22×D20, S3×D20 [×8], C2×C3⋊D20 [×2], C6×D20, S3×C2×C20, C2×D60, C22×S3×D5 [×2], C2×S3×D20
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D5, D6 [×7], C2×D4 [×6], C24, D10 [×7], C22×S3 [×7], C22×D4, D20 [×4], C22×D5 [×7], S3×D4 [×2], S3×C23, S3×D5, C2×D20 [×6], C23×D5, C2×S3×D4, C2×S3×D5 [×3], C22×D20, S3×D20 [×2], C22×S3×D5, C2×S3×D20

Smallest permutation representation of C2×S3×D20
On 120 points
Generators in S120
(1 104)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 112)(10 113)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 101)(19 102)(20 103)(21 63)(22 64)(23 65)(24 66)(25 67)(26 68)(27 69)(28 70)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 61)(40 62)(41 98)(42 99)(43 100)(44 81)(45 82)(46 83)(47 84)(48 85)(49 86)(50 87)(51 88)(52 89)(53 90)(54 91)(55 92)(56 93)(57 94)(58 95)(59 96)(60 97)
(1 78 54)(2 79 55)(3 80 56)(4 61 57)(5 62 58)(6 63 59)(7 64 60)(8 65 41)(9 66 42)(10 67 43)(11 68 44)(12 69 45)(13 70 46)(14 71 47)(15 72 48)(16 73 49)(17 74 50)(18 75 51)(19 76 52)(20 77 53)(21 96 109)(22 97 110)(23 98 111)(24 99 112)(25 100 113)(26 81 114)(27 82 115)(28 83 116)(29 84 117)(30 85 118)(31 86 119)(32 87 120)(33 88 101)(34 89 102)(35 90 103)(36 91 104)(37 92 105)(38 93 106)(39 94 107)(40 95 108)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 86)(22 87)(23 88)(24 89)(25 90)(26 91)(27 92)(28 93)(29 94)(30 95)(31 96)(32 97)(33 98)(34 99)(35 100)(36 81)(37 82)(38 83)(39 84)(40 85)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 61)(48 62)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(57 71)(58 72)(59 73)(60 74)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 46)(42 45)(43 44)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(75 80)(76 79)(77 78)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(101 106)(102 105)(103 104)(107 120)(108 119)(109 118)(110 117)(111 116)(112 115)(113 114)

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

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

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

78 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 15A 15B 20A ··· 20H 20I ··· 20P 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 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 3 3 3 3 10 10 10 10 30 30 30 30 2 2 2 6 6 2 2 2 2 2 20 20 20 20 2 ··· 2 6 ··· 6 4 4 4 4 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

78 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 D10 D10 D10 D10 D20 S3×D4 S3×D5 C2×S3×D5 C2×S3×D5 S3×D20 kernel C2×S3×D20 S3×D20 C2×C3⋊D20 C6×D20 S3×C2×C20 C2×D60 C22×S3×D5 C2×D20 S3×C10 S3×C2×C4 D20 C2×C20 C22×D5 C4×S3 C2×Dic3 C2×C12 C22×S3 D6 C10 C2×C4 C4 C22 C2 # reps 1 8 2 1 1 1 2 1 4 2 4 1 2 8 2 2 2 16 2 2 4 2 8

Matrix representation of C2×S3×D20 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 0 0 0 0 1 60
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 34 32 0 0 0 0 2 36 0 0 0 0 0 0 27 29 0 0 0 0 59 25 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 25 32 0 0 0 0 11 36 0 0 0 0 0 0 36 29 0 0 0 0 50 25 0 0 0 0 0 0 60 0 0 0 0 0 0 60

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[34,2,0,0,0,0,32,36,0,0,0,0,0,0,27,59,0,0,0,0,29,25,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[25,11,0,0,0,0,32,36,0,0,0,0,0,0,36,50,0,0,0,0,29,25,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;

C2×S3×D20 in GAP, Magma, Sage, TeX

C_2\times S_3\times D_{20}
% in TeX

G:=Group("C2xS3xD20");
// GroupNames label

G:=SmallGroup(480,1088);
// by ID

G=gap.SmallGroup(480,1088);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^20=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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