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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 [×6], C2 [×4], C3, C4 [×4], C22, C22 [×10], C22 [×12], C6, C6 [×6], C6 [×4], C2×C4 [×8], C23, C23 [×6], C23 [×4], C9, Dic3 [×4], C2×C6, C2×C6 [×10], C2×C6 [×12], C22⋊C4 [×4], C22×C4 [×2], C24, C18, C18 [×6], C18 [×4], C2×Dic3 [×8], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C22⋊C4, Dic9 [×4], C2×C18, C2×C18 [×10], C2×C18 [×12], C6.D4 [×4], C22×Dic3 [×2], C23×C6, C2×Dic9 [×4], C2×Dic9 [×4], C22×C18, C22×C18 [×6], C22×C18 [×4], C2×C6.D4, C18.D4 [×4], C22×Dic9 [×2], C23×C18, C2×C18.D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], D9, C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C2×C22⋊C4, Dic9 [×4], D18 [×3], C6.D4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C2×Dic9 [×6], C9⋊D4 [×4], C22×D9, C2×C6.D4, C18.D4 [×4], C22×Dic9, C2×C9⋊D4 [×2], C2×C18.D4

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

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

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

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

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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