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G = C2×Dic18order 144 = 24·32

Direct product of C2 and Dic18

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×Dic18, C18⋊Q8, C4.11D18, C12.43D6, C6.5Dic6, C18.1C23, C22.8D18, C36.11C22, Dic9.1C22, C91(C2×Q8), (C2×C4).4D9, C3.(C2×Dic6), (C2×C36).3C2, (C2×C12).6S3, (C2×C6).24D6, C2.3(C22×D9), C6.19(C22×S3), (C2×C18).8C22, (C2×Dic9).3C2, SmallGroup(144,37)

Series: Derived Chief Lower central Upper central

C1C18 — C2×Dic18
C1C3C9C18Dic9C2×Dic9 — C2×Dic18
C9C18 — C2×Dic18
C1C22C2×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 [×2], C3, C4 [×2], C4 [×4], C22, C6, C6 [×2], C2×C4, C2×C4 [×2], Q8 [×4], C9, Dic3 [×4], C12 [×2], C2×C6, C2×Q8, C18, C18 [×2], Dic6 [×4], C2×Dic3 [×2], C2×C12, Dic9 [×4], C36 [×2], C2×C18, C2×Dic6, Dic18 [×4], C2×Dic9 [×2], C2×C36, C2×Dic18
Quotients: C1, C2 [×7], C22 [×7], S3, Q8 [×2], C23, D6 [×3], C2×Q8, D9, Dic6 [×2], C22×S3, D18 [×3], C2×Dic6, Dic18 [×2], C22×D9, C2×Dic18

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

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

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

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

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 2A2B2C 3 4A4B4C4D4E4F6A6B6C9A9B9C12A12B12C12D18A···18I36A···36L
order122234444446669991212121218···1836···36
size11112221818181822222222222···22···2

42 irreducible representations

dim1111222222222
type+++++-+++-++-
imageC1C2C2C2S3Q8D6D6D9Dic6D18D18Dic18
kernelC2×Dic18Dic18C2×Dic9C2×C36C2×C12C18C12C2×C6C2×C4C6C4C22C2
# reps14211221346312

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

3600
010
001
,
100
0429
0812
,
100
0836
02829
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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