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## G = C2×S3×Dic5order 240 = 24·3·5

### Direct product of C2, S3 and Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C2×S3×Dic5
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — S3×Dic5 — C2×S3×Dic5
 Lower central C15 — C2×S3×Dic5
 Upper central C1 — C22

Generators and relations for C2×S3×Dic5
G = < a,b,c,d,e | a2=b3=c2=d10=1, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 336 in 108 conjugacy classes, 56 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C23, C10, C10, C10, Dic3, C12, D6, C2×C6, C15, C22×C4, Dic5, Dic5, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C30, C2×Dic5, C2×Dic5, C22×C10, S3×C2×C4, C3×Dic5, Dic15, S3×C10, C2×C30, C22×Dic5, S3×Dic5, C6×Dic5, C2×Dic15, S3×C2×C10, C2×S3×Dic5
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, Dic5, D10, C4×S3, C22×S3, C2×Dic5, C22×D5, S3×C2×C4, S3×D5, C22×Dic5, S3×Dic5, C2×S3×D5, C2×S3×Dic5

Smallest permutation representation of C2×S3×Dic5
On 120 points
Generators in S120
(1 59)(2 60)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 91)(21 48)(22 49)(23 50)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 68)(32 69)(33 70)(34 61)(35 62)(36 63)(37 64)(38 65)(39 66)(40 67)(71 101)(72 102)(73 103)(74 104)(75 105)(76 106)(77 107)(78 108)(79 109)(80 110)(81 111)(82 112)(83 113)(84 114)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)
(1 21 33)(2 22 34)(3 23 35)(4 24 36)(5 25 37)(6 26 38)(7 27 39)(8 28 40)(9 29 31)(10 30 32)(11 111 107)(12 112 108)(13 113 109)(14 114 110)(15 115 101)(16 116 102)(17 117 103)(18 118 104)(19 119 105)(20 120 106)(41 63 52)(42 64 53)(43 65 54)(44 66 55)(45 67 56)(46 68 57)(47 69 58)(48 70 59)(49 61 60)(50 62 51)(71 96 85)(72 97 86)(73 98 87)(74 99 88)(75 100 89)(76 91 90)(77 92 81)(78 93 82)(79 94 83)(80 95 84)
(1 54)(2 55)(3 56)(4 57)(5 58)(6 59)(7 60)(8 51)(9 52)(10 53)(11 72)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 80)(20 71)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 61)(28 62)(29 63)(30 64)(31 41)(32 42)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 49)(40 50)(81 116)(82 117)(83 118)(84 119)(85 120)(86 111)(87 112)(88 113)(89 114)(90 115)(91 101)(92 102)(93 103)(94 104)(95 105)(96 106)(97 107)(98 108)(99 109)(100 110)
(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 86 6 81)(2 85 7 90)(3 84 8 89)(4 83 9 88)(5 82 10 87)(11 70 16 65)(12 69 17 64)(13 68 18 63)(14 67 19 62)(15 66 20 61)(21 72 26 77)(22 71 27 76)(23 80 28 75)(24 79 29 74)(25 78 30 73)(31 99 36 94)(32 98 37 93)(33 97 38 92)(34 96 39 91)(35 95 40 100)(41 109 46 104)(42 108 47 103)(43 107 48 102)(44 106 49 101)(45 105 50 110)(51 114 56 119)(52 113 57 118)(53 112 58 117)(54 111 59 116)(55 120 60 115)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 6A 6B 6C 10A ··· 10F 10G ··· 10N 12A 12B 12C 12D 15A 15B 30A ··· 30F order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 5 5 6 6 6 10 ··· 10 10 ··· 10 12 12 12 12 15 15 30 ··· 30 size 1 1 1 1 3 3 3 3 2 5 5 5 5 15 15 15 15 2 2 2 2 2 2 ··· 2 6 ··· 6 10 10 10 10 4 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + - + + + - + image C1 C2 C2 C2 C2 C4 S3 D5 D6 D6 Dic5 D10 D10 C4×S3 S3×D5 S3×Dic5 C2×S3×D5 kernel C2×S3×Dic5 S3×Dic5 C6×Dic5 C2×Dic15 S3×C2×C10 S3×C10 C2×Dic5 C22×S3 Dic5 C2×C10 D6 D6 C2×C6 C10 C22 C2 C2 # reps 1 4 1 1 1 8 1 2 2 1 8 4 2 4 2 4 2

Matrix representation of C2×S3×Dic5 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 1 15 0 0 12 59 0 0 0 0 1 0 0 0 0 1
,
 1 15 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 60 0 0 0 0 60 0 0 0 0 43 1 0 0 60 0
,
 50 0 0 0 0 50 0 0 0 0 16 9 0 0 53 45
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,12,0,0,15,59,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,15,60,0,0,0,0,60,0,0,0,0,60],[60,0,0,0,0,60,0,0,0,0,43,60,0,0,1,0],[50,0,0,0,0,50,0,0,0,0,16,53,0,0,9,45] >;

C2×S3×Dic5 in GAP, Magma, Sage, TeX

C_2\times S_3\times {\rm Dic}_5
% in TeX

G:=Group("C2xS3xDic5");
// GroupNames label

G:=SmallGroup(240,142);
// by ID

G=gap.SmallGroup(240,142);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,48,490,6917]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^10=1,e^2=d^5,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^-1=d^-1>;
// generators/relations

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