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## G = C2×Dic18order 144 = 24·32

### Direct product of C2 and Dic18

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C2×Dic18
 Chief series C1 — C3 — C9 — C18 — Dic9 — C2×Dic9 — C2×Dic18
 Lower central C9 — C18 — C2×Dic18
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×Dic18
G = < a,b,c | a2=b36=1, c2=b18, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 167 in 57 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C2×C4, C2×C4, Q8, C9, Dic3, C12, C2×C6, C2×Q8, C18, C18, Dic6, C2×Dic3, C2×C12, Dic9, C36, C2×C18, C2×Dic6, Dic18, C2×Dic9, C2×C36, C2×Dic18
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, D9, Dic6, C22×S3, D18, C2×Dic6, Dic18, C22×D9, C2×Dic18

Smallest permutation representation of C2×Dic18
Regular action on 144 points
Generators in S144
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 71)(18 72)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(73 113)(74 114)(75 115)(76 116)(77 117)(78 118)(79 119)(80 120)(81 121)(82 122)(83 123)(84 124)(85 125)(86 126)(87 127)(88 128)(89 129)(90 130)(91 131)(92 132)(93 133)(94 134)(95 135)(96 136)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 109)(106 110)(107 111)(108 112)
(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)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)
(1 90 19 108)(2 89 20 107)(3 88 21 106)(4 87 22 105)(5 86 23 104)(6 85 24 103)(7 84 25 102)(8 83 26 101)(9 82 27 100)(10 81 28 99)(11 80 29 98)(12 79 30 97)(13 78 31 96)(14 77 32 95)(15 76 33 94)(16 75 34 93)(17 74 35 92)(18 73 36 91)(37 112 55 130)(38 111 56 129)(39 110 57 128)(40 109 58 127)(41 144 59 126)(42 143 60 125)(43 142 61 124)(44 141 62 123)(45 140 63 122)(46 139 64 121)(47 138 65 120)(48 137 66 119)(49 136 67 118)(50 135 68 117)(51 134 69 116)(52 133 70 115)(53 132 71 114)(54 131 72 113)

G:=sub<Sym(144)| (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(73,113)(74,114)(75,115)(76,116)(77,117)(78,118)(79,119)(80,120)(81,121)(82,122)(83,123)(84,124)(85,125)(86,126)(87,127)(88,128)(89,129)(90,130)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,109)(106,110)(107,111)(108,112), (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)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,90,19,108)(2,89,20,107)(3,88,21,106)(4,87,22,105)(5,86,23,104)(6,85,24,103)(7,84,25,102)(8,83,26,101)(9,82,27,100)(10,81,28,99)(11,80,29,98)(12,79,30,97)(13,78,31,96)(14,77,32,95)(15,76,33,94)(16,75,34,93)(17,74,35,92)(18,73,36,91)(37,112,55,130)(38,111,56,129)(39,110,57,128)(40,109,58,127)(41,144,59,126)(42,143,60,125)(43,142,61,124)(44,141,62,123)(45,140,63,122)(46,139,64,121)(47,138,65,120)(48,137,66,119)(49,136,67,118)(50,135,68,117)(51,134,69,116)(52,133,70,115)(53,132,71,114)(54,131,72,113)>;

G:=Group( (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,61)(8,62)(9,63)(10,64)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,71)(18,72)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,49)(32,50)(33,51)(34,52)(35,53)(36,54)(73,113)(74,114)(75,115)(76,116)(77,117)(78,118)(79,119)(80,120)(81,121)(82,122)(83,123)(84,124)(85,125)(86,126)(87,127)(88,128)(89,129)(90,130)(91,131)(92,132)(93,133)(94,134)(95,135)(96,136)(97,137)(98,138)(99,139)(100,140)(101,141)(102,142)(103,143)(104,144)(105,109)(106,110)(107,111)(108,112), (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)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144), (1,90,19,108)(2,89,20,107)(3,88,21,106)(4,87,22,105)(5,86,23,104)(6,85,24,103)(7,84,25,102)(8,83,26,101)(9,82,27,100)(10,81,28,99)(11,80,29,98)(12,79,30,97)(13,78,31,96)(14,77,32,95)(15,76,33,94)(16,75,34,93)(17,74,35,92)(18,73,36,91)(37,112,55,130)(38,111,56,129)(39,110,57,128)(40,109,58,127)(41,144,59,126)(42,143,60,125)(43,142,61,124)(44,141,62,123)(45,140,63,122)(46,139,64,121)(47,138,65,120)(48,137,66,119)(49,136,67,118)(50,135,68,117)(51,134,69,116)(52,133,70,115)(53,132,71,114)(54,131,72,113) );

G=PermutationGroup([[(1,55),(2,56),(3,57),(4,58),(5,59),(6,60),(7,61),(8,62),(9,63),(10,64),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,71),(18,72),(19,37),(20,38),(21,39),(22,40),(23,41),(24,42),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,49),(32,50),(33,51),(34,52),(35,53),(36,54),(73,113),(74,114),(75,115),(76,116),(77,117),(78,118),(79,119),(80,120),(81,121),(82,122),(83,123),(84,124),(85,125),(86,126),(87,127),(88,128),(89,129),(90,130),(91,131),(92,132),(93,133),(94,134),(95,135),(96,136),(97,137),(98,138),(99,139),(100,140),(101,141),(102,142),(103,143),(104,144),(105,109),(106,110),(107,111),(108,112)], [(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),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144)], [(1,90,19,108),(2,89,20,107),(3,88,21,106),(4,87,22,105),(5,86,23,104),(6,85,24,103),(7,84,25,102),(8,83,26,101),(9,82,27,100),(10,81,28,99),(11,80,29,98),(12,79,30,97),(13,78,31,96),(14,77,32,95),(15,76,33,94),(16,75,34,93),(17,74,35,92),(18,73,36,91),(37,112,55,130),(38,111,56,129),(39,110,57,128),(40,109,58,127),(41,144,59,126),(42,143,60,125),(43,142,61,124),(44,141,62,123),(45,140,63,122),(46,139,64,121),(47,138,65,120),(48,137,66,119),(49,136,67,118),(50,135,68,117),(51,134,69,116),(52,133,70,115),(53,132,71,114),(54,131,72,113)]])

C2×Dic18 is a maximal subgroup of
C18.Q16  C36.45D4  C4.D36  C362Q8  C427D9  C222Dic18  Dic9.D4  Dic93Q8  C36⋊Q8  D18⋊Q8  D182Q8  C8.D18  C36.49D4  C36.17D4  Dic9⋊Q8  D4.D18  C2×Q8×D9  D4.10D18
C2×Dic18 is a maximal quotient of
C362Q8  C36.6Q8  C222Dic18  C36⋊Q8  C36.3Q8  C36.49D4

42 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 9A 9B 9C 12A 12B 12C 12D 18A ··· 18I 36A ··· 36L order 1 2 2 2 3 4 4 4 4 4 4 6 6 6 9 9 9 12 12 12 12 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 18 18 18 18 2 2 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + - + + + - + + - image C1 C2 C2 C2 S3 Q8 D6 D6 D9 Dic6 D18 D18 Dic18 kernel C2×Dic18 Dic18 C2×Dic9 C2×C36 C2×C12 C18 C12 C2×C6 C2×C4 C6 C4 C22 C2 # reps 1 4 2 1 1 2 2 1 3 4 6 3 12

Matrix representation of C2×Dic18 in GL3(𝔽37) generated by

 36 0 0 0 1 0 0 0 1
,
 1 0 0 0 4 29 0 8 12
,
 1 0 0 0 8 36 0 28 29
G:=sub<GL(3,GF(37))| [36,0,0,0,1,0,0,0,1],[1,0,0,0,4,8,0,29,12],[1,0,0,0,8,28,0,36,29] >;

C2×Dic18 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{18}
% in TeX

G:=Group("C2xDic18");
// GroupNames label

G:=SmallGroup(144,37);
// by ID

G=gap.SmallGroup(144,37);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-3,48,218,50,2404,208,3461]);
// Polycyclic

G:=Group<a,b,c|a^2=b^36=1,c^2=b^18,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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