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## G = C2×C18.D4order 288 = 25·32

### Direct product of C2 and C18.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C2×C18.D4
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×Dic9 — C22×Dic9 — C2×C18.D4
 Lower central C9 — C18 — C2×C18.D4
 Upper central C1 — C23 — C24

Generators and relations for C2×C18.D4
G = < a,b,c,d | a2=b18=c4=1, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b9c-1 >

Subgroups: 552 in 198 conjugacy classes, 92 normal (16 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C23, C23, C9, Dic3, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C24, C18, C18, C18, C2×Dic3, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, Dic9, C2×C18, C2×C18, C2×C18, C6.D4, C22×Dic3, C23×C6, C2×Dic9, C2×Dic9, C22×C18, C22×C18, C22×C18, C2×C6.D4, C18.D4, C22×Dic9, C23×C18, C2×C18.D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, D9, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, Dic9, D18, C6.D4, C22×Dic3, C2×C3⋊D4, C2×Dic9, C9⋊D4, C22×D9, C2×C6.D4, C18.D4, C22×Dic9, C2×C9⋊D4, C2×C18.D4

Smallest permutation representation of C2×C18.D4
On 144 points
Generators in S144
(1 46)(2 47)(3 48)(4 49)(5 50)(6 51)(7 52)(8 53)(9 54)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 45)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 55)(34 56)(35 57)(36 58)(73 131)(74 132)(75 133)(76 134)(77 135)(78 136)(79 137)(80 138)(81 139)(82 140)(83 141)(84 142)(85 143)(86 144)(87 127)(88 128)(89 129)(90 130)(91 118)(92 119)(93 120)(94 121)(95 122)(96 123)(97 124)(98 125)(99 126)(100 109)(101 110)(102 111)(103 112)(104 113)(105 114)(106 115)(107 116)(108 117)
(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)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126)(127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144)
(1 82 22 109)(2 81 23 126)(3 80 24 125)(4 79 25 124)(5 78 26 123)(6 77 27 122)(7 76 28 121)(8 75 29 120)(9 74 30 119)(10 73 31 118)(11 90 32 117)(12 89 33 116)(13 88 34 115)(14 87 35 114)(15 86 36 113)(16 85 19 112)(17 84 20 111)(18 83 21 110)(37 131 71 91)(38 130 72 108)(39 129 55 107)(40 128 56 106)(41 127 57 105)(42 144 58 104)(43 143 59 103)(44 142 60 102)(45 141 61 101)(46 140 62 100)(47 139 63 99)(48 138 64 98)(49 137 65 97)(50 136 66 96)(51 135 67 95)(52 134 68 94)(53 133 69 93)(54 132 70 92)
(1 91 10 100)(2 108 11 99)(3 107 12 98)(4 106 13 97)(5 105 14 96)(6 104 15 95)(7 103 16 94)(8 102 17 93)(9 101 18 92)(19 134 28 143)(20 133 29 142)(21 132 30 141)(22 131 31 140)(23 130 32 139)(24 129 33 138)(25 128 34 137)(26 127 35 136)(27 144 36 135)(37 109 46 118)(38 126 47 117)(39 125 48 116)(40 124 49 115)(41 123 50 114)(42 122 51 113)(43 121 52 112)(44 120 53 111)(45 119 54 110)(55 80 64 89)(56 79 65 88)(57 78 66 87)(58 77 67 86)(59 76 68 85)(60 75 69 84)(61 74 70 83)(62 73 71 82)(63 90 72 81)

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A ··· 4H 6A ··· 6O 9A 9B 9C 18A ··· 18AS order 1 2 ··· 2 2 2 2 2 3 4 ··· 4 6 ··· 6 9 9 9 18 ··· 18 size 1 1 ··· 1 2 2 2 2 2 18 ··· 18 2 ··· 2 2 2 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + - + + - + image C1 C2 C2 C2 C4 S3 D4 Dic3 D6 D9 C3⋊D4 Dic9 D18 C9⋊D4 kernel C2×C18.D4 C18.D4 C22×Dic9 C23×C18 C22×C18 C23×C6 C2×C18 C22×C6 C22×C6 C24 C2×C6 C23 C23 C22 # reps 1 4 2 1 8 1 4 4 3 3 8 12 9 24

Matrix representation of C2×C18.D4 in GL5(𝔽37)

 36 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 26 0 0 0 0 0 10 0 0 0 0 0 30 0 0 0 0 0 21
,
 1 0 0 0 0 0 0 1 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 36 0

G:=sub<GL(5,GF(37))| [36,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,26,0,0,0,0,0,10,0,0,0,0,0,30,0,0,0,0,0,21],[1,0,0,0,0,0,0,36,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,36,0,0,0,1,0] >;

C2×C18.D4 in GAP, Magma, Sage, TeX

C_2\times C_{18}.D_4
% in TeX

G:=Group("C2xC18.D4");
// GroupNames label

G:=SmallGroup(288,162);
// by ID

G=gap.SmallGroup(288,162);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,422,6725,292,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^18=c^4=1,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^9*c^-1>;
// generators/relations

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