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## G = C2×D18⋊C4order 288 = 25·32

### Direct product of C2 and D18⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C2×D18⋊C4
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×D9 — C23×D9 — C2×D18⋊C4
 Lower central C9 — C18 — C2×D18⋊C4
 Upper central C1 — C23 — C22×C4

Generators and relations for C2×D18⋊C4
G = < a,b,c,d | a2=b18=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=b9c >

Subgroups: 952 in 198 conjugacy classes, 76 normal (24 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C23, C9, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C24, D9, C18, C18, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C2×C22⋊C4, Dic9, C36, D18, D18, C2×C18, C2×C18, D6⋊C4, C22×Dic3, C22×C12, S3×C23, C2×Dic9, C2×Dic9, C2×C36, C2×C36, C22×D9, C22×D9, C22×C18, C2×D6⋊C4, D18⋊C4, C22×Dic9, C22×C36, C23×D9, C2×D18⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, D9, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, D18, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C4×D9, D36, C9⋊D4, C22×D9, C2×D6⋊C4, D18⋊C4, C2×C4×D9, C2×D36, C2×C9⋊D4, C2×D18⋊C4

Smallest permutation representation of C2×D18⋊C4
On 144 points
Generators in S144
(1 108)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 97)(9 98)(10 99)(11 100)(12 101)(13 102)(14 103)(15 104)(16 105)(17 106)(18 107)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 84)(26 85)(27 86)(28 87)(29 88)(30 89)(31 90)(32 73)(33 74)(34 75)(35 76)(36 77)(37 143)(38 144)(39 127)(40 128)(41 129)(42 130)(43 131)(44 132)(45 133)(46 134)(47 135)(48 136)(49 137)(50 138)(51 139)(52 140)(53 141)(54 142)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)(61 115)(62 116)(63 117)(64 118)(65 119)(66 120)(67 121)(68 122)(69 123)(70 124)(71 125)(72 126)
(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)(2 89)(3 88)(4 87)(5 86)(6 85)(7 84)(8 83)(9 82)(10 81)(11 80)(12 79)(13 78)(14 77)(15 76)(16 75)(17 74)(18 73)(19 102)(20 101)(21 100)(22 99)(23 98)(24 97)(25 96)(26 95)(27 94)(28 93)(29 92)(30 91)(31 108)(32 107)(33 106)(34 105)(35 104)(36 103)(37 126)(38 125)(39 124)(40 123)(41 122)(42 121)(43 120)(44 119)(45 118)(46 117)(47 116)(48 115)(49 114)(50 113)(51 112)(52 111)(53 110)(54 109)(55 142)(56 141)(57 140)(58 139)(59 138)(60 137)(61 136)(62 135)(63 134)(64 133)(65 132)(66 131)(67 130)(68 129)(69 128)(70 127)(71 144)(72 143)
(1 144 32 117)(2 127 33 118)(3 128 34 119)(4 129 35 120)(5 130 36 121)(6 131 19 122)(7 132 20 123)(8 133 21 124)(9 134 22 125)(10 135 23 126)(11 136 24 109)(12 137 25 110)(13 138 26 111)(14 139 27 112)(15 140 28 113)(16 141 29 114)(17 142 30 115)(18 143 31 116)(37 90 62 107)(38 73 63 108)(39 74 64 91)(40 75 65 92)(41 76 66 93)(42 77 67 94)(43 78 68 95)(44 79 69 96)(45 80 70 97)(46 81 71 98)(47 82 72 99)(48 83 55 100)(49 84 56 101)(50 85 57 102)(51 86 58 103)(52 87 59 104)(53 88 60 105)(54 89 61 106)

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6G 9A 9B 9C 12A ··· 12H 18A ··· 18U 36A ··· 36X order 1 2 ··· 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 ··· 6 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 ··· 1 18 18 18 18 2 2 2 2 2 18 18 18 18 2 ··· 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D4 D6 D6 D9 C4×S3 D12 C3⋊D4 D18 D18 C4×D9 D36 C9⋊D4 kernel C2×D18⋊C4 D18⋊C4 C22×Dic9 C22×C36 C23×D9 C22×D9 C22×C12 C2×C18 C2×C12 C22×C6 C22×C4 C2×C6 C2×C6 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 4 1 1 1 8 1 4 2 1 3 4 4 4 6 3 12 12 12

Matrix representation of C2×D18⋊C4 in GL6(𝔽37)

 36 0 0 0 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 20 31 0 0 0 0 6 26
,
 1 0 0 0 0 0 0 36 0 0 0 0 0 0 1 0 0 0 0 0 0 36 0 0 0 0 0 0 17 6 0 0 0 0 26 20
,
 0 36 0 0 0 0 36 0 0 0 0 0 0 0 0 36 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,20,6,0,0,0,0,31,26],[1,0,0,0,0,0,0,36,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,17,26,0,0,0,0,6,20],[0,36,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C2×D18⋊C4 in GAP, Magma, Sage, TeX

C_2\times D_{18}\rtimes C_4
% in TeX

G:=Group("C2xD18:C4");
// GroupNames label

G:=SmallGroup(288,137);
// by ID

G=gap.SmallGroup(288,137);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,58,6725,292,9414]);
// Polycyclic

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

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