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## G = D4×C2×C18order 288 = 25·32

### Direct product of C2×C18 and D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C2×C18
 Chief series C1 — C3 — C6 — C18 — C2×C18 — D4×C9 — D4×C18 — D4×C2×C18
 Lower central C1 — C2 — D4×C2×C18
 Upper central C1 — C22×C18 — D4×C2×C18

Generators and relations for D4×C2×C18
G = < a,b,c,d | a2=b18=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 474 in 354 conjugacy classes, 234 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C2×C4, D4, C23, C23, C23, C9, C12, C2×C6, C2×C6, C22×C4, C2×D4, C24, C18, C18, C18, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C22×D4, C36, C2×C18, C2×C18, C22×C12, C6×D4, C23×C6, C2×C36, D4×C9, C22×C18, C22×C18, C22×C18, D4×C2×C6, C22×C36, D4×C18, C23×C18, D4×C2×C18
Quotients: C1, C2, C3, C22, C6, D4, C23, C9, C2×C6, C2×D4, C24, C18, C3×D4, C22×C6, C22×D4, C2×C18, C6×D4, C23×C6, D4×C9, C22×C18, D4×C2×C6, D4×C18, C23×C18, D4×C2×C18

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

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

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

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

180 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3A 3B 4A 4B 4C 4D 6A ··· 6N 6O ··· 6AD 9A ··· 9F 12A ··· 12H 18A ··· 18AP 18AQ ··· 18CL 36A ··· 36X order 1 2 ··· 2 2 ··· 2 3 3 4 4 4 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 size 1 1 ··· 1 2 ··· 2 1 1 2 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 type + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 C9 C18 C18 C18 D4 C3×D4 D4×C9 kernel D4×C2×C18 C22×C36 D4×C18 C23×C18 D4×C2×C6 C22×C12 C6×D4 C23×C6 C22×D4 C22×C4 C2×D4 C24 C2×C18 C2×C6 C22 # reps 1 1 12 2 2 2 24 4 6 6 72 12 4 8 24

Matrix representation of D4×C2×C18 in GL4(𝔽37) generated by

 36 0 0 0 0 36 0 0 0 0 1 0 0 0 0 1
,
 11 0 0 0 0 10 0 0 0 0 33 0 0 0 0 33
,
 36 0 0 0 0 36 0 0 0 0 36 2 0 0 36 1
,
 36 0 0 0 0 1 0 0 0 0 36 0 0 0 36 1
G:=sub<GL(4,GF(37))| [36,0,0,0,0,36,0,0,0,0,1,0,0,0,0,1],[11,0,0,0,0,10,0,0,0,0,33,0,0,0,0,33],[36,0,0,0,0,36,0,0,0,0,36,36,0,0,2,1],[36,0,0,0,0,1,0,0,0,0,36,36,0,0,0,1] >;

D4×C2×C18 in GAP, Magma, Sage, TeX

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

G:=Group("D4xC2xC18");
// GroupNames label

G:=SmallGroup(288,368);
// by ID

G=gap.SmallGroup(288,368);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,701,242]);
// Polycyclic

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

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