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## G = C2×D5×Dic3order 240 = 24·3·5

### Direct product of C2, D5 and Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C15 — C2×D5×Dic3
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×Dic3 — C2×D5×Dic3
 Lower central C15 — C2×D5×Dic3
 Upper central C1 — C22

Generators and relations for C2×D5×Dic3
G = < a,b,c,d,e | a2=b5=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 368 in 108 conjugacy classes, 56 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, C6, C6, C6, C2×C4, C23, D5, C10, C10, Dic3, Dic3, C2×C6, C2×C6, C15, C22×C4, Dic5, C20, D10, C2×C10, C2×Dic3, C2×Dic3, C22×C6, C3×D5, C30, C30, C4×D5, C2×Dic5, C2×C20, C22×D5, C22×Dic3, C5×Dic3, Dic15, C6×D5, C2×C30, C2×C4×D5, D5×Dic3, C10×Dic3, C2×Dic15, D5×C2×C6, C2×D5×Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, Dic3, D6, C22×C4, D10, C2×Dic3, C22×S3, C4×D5, C22×D5, C22×Dic3, S3×D5, C2×C4×D5, D5×Dic3, C2×S3×D5, C2×D5×Dic3

Smallest permutation representation of C2×D5×Dic3
On 120 points
Generators in S120
(1 58)(2 59)(3 60)(4 55)(5 56)(6 57)(7 112)(8 113)(9 114)(10 109)(11 110)(12 111)(13 108)(14 103)(15 104)(16 105)(17 106)(18 107)(19 98)(20 99)(21 100)(22 101)(23 102)(24 97)(25 94)(26 95)(27 96)(28 91)(29 92)(30 93)(31 72)(32 67)(33 68)(34 69)(35 70)(36 71)(37 62)(38 63)(39 64)(40 65)(41 66)(42 61)(43 84)(44 79)(45 80)(46 81)(47 82)(48 83)(49 74)(50 75)(51 76)(52 77)(53 78)(54 73)(85 115)(86 116)(87 117)(88 118)(89 119)(90 120)
(1 39 51 48 31)(2 40 52 43 32)(3 41 53 44 33)(4 42 54 45 34)(5 37 49 46 35)(6 38 50 47 36)(7 24 115 26 14)(8 19 116 27 15)(9 20 117 28 16)(10 21 118 29 17)(11 22 119 30 18)(12 23 120 25 13)(55 61 73 80 69)(56 62 74 81 70)(57 63 75 82 71)(58 64 76 83 72)(59 65 77 84 67)(60 66 78 79 68)(85 95 103 112 97)(86 96 104 113 98)(87 91 105 114 99)(88 92 106 109 100)(89 93 107 110 101)(90 94 108 111 102)
(1 69)(2 70)(3 71)(4 72)(5 67)(6 68)(7 92)(8 93)(9 94)(10 95)(11 96)(12 91)(13 105)(14 106)(15 107)(16 108)(17 103)(18 104)(19 89)(20 90)(21 85)(22 86)(23 87)(24 88)(25 114)(26 109)(27 110)(28 111)(29 112)(30 113)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 84)(38 79)(39 80)(40 81)(41 82)(42 83)(43 62)(44 63)(45 64)(46 65)(47 66)(48 61)(49 77)(50 78)(51 73)(52 74)(53 75)(54 76)(97 118)(98 119)(99 120)(100 115)(101 116)(102 117)
(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)
(1 119 4 116)(2 118 5 115)(3 117 6 120)(7 43 10 46)(8 48 11 45)(9 47 12 44)(13 53 16 50)(14 52 17 49)(15 51 18 54)(19 31 22 34)(20 36 23 33)(21 35 24 32)(25 41 28 38)(26 40 29 37)(27 39 30 42)(55 86 58 89)(56 85 59 88)(57 90 60 87)(61 96 64 93)(62 95 65 92)(63 94 66 91)(67 100 70 97)(68 99 71 102)(69 98 72 101)(73 104 76 107)(74 103 77 106)(75 108 78 105)(79 114 82 111)(80 113 83 110)(81 112 84 109)

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

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

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

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 6D 6E 6F 6G 10A ··· 10F 15A 15B 20A ··· 20H 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 6 6 6 6 10 ··· 10 15 15 20 ··· 20 30 ··· 30 size 1 1 1 1 5 5 5 5 2 3 3 3 3 15 15 15 15 2 2 2 2 2 10 10 10 10 2 ··· 2 4 4 6 ··· 6 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 Dic3 D6 D6 D10 D10 C4×D5 S3×D5 D5×Dic3 C2×S3×D5 kernel C2×D5×Dic3 D5×Dic3 C10×Dic3 C2×Dic15 D5×C2×C6 C6×D5 C22×D5 C2×Dic3 D10 D10 C2×C10 Dic3 C2×C6 C6 C22 C2 C2 # reps 1 4 1 1 1 8 1 2 4 2 1 4 2 8 2 4 2

Matrix representation of C2×D5×Dic3 in GL5(𝔽61)

 60 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 60 1 0 0 0 42 18
,
 60 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 19 60
,
 60 0 0 0 0 0 59 46 0 0 0 49 1 0 0 0 0 0 60 0 0 0 0 0 60
,
 11 0 0 0 0 0 4 48 0 0 0 34 57 0 0 0 0 0 11 0 0 0 0 0 11

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,42,0,0,0,1,18],[60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,19,0,0,0,0,60],[60,0,0,0,0,0,59,49,0,0,0,46,1,0,0,0,0,0,60,0,0,0,0,0,60],[11,0,0,0,0,0,4,34,0,0,0,48,57,0,0,0,0,0,11,0,0,0,0,0,11] >;

C2×D5×Dic3 in GAP, Magma, Sage, TeX

C_2\times D_5\times {\rm Dic}_3
% in TeX

G:=Group("C2xD5xDic3");
// GroupNames label

G:=SmallGroup(240,139);
// by ID

G=gap.SmallGroup(240,139);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,55,490,6917]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^6=1,e^2=d^3,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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