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## G = C4○D4×C18order 288 = 25·32

### Direct product of C18 and C4○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4×C18
 Chief series C1 — C3 — C6 — C18 — C2×C18 — D4×C9 — C9×C4○D4 — C4○D4×C18
 Lower central C1 — C2 — C4○D4×C18
 Upper central C1 — C2×C36 — C4○D4×C18

Generators and relations for C4○D4×C18
G = < a,b,c,d | a18=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 282 in 246 conjugacy classes, 210 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C9, C12, C2×C6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C18, C18, C18, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4○D4, C36, C2×C18, C2×C18, C2×C18, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C2×C36, C2×C36, D4×C9, Q8×C9, C22×C18, C6×C4○D4, C22×C36, D4×C18, Q8×C18, C9×C4○D4, C4○D4×C18
Quotients: C1, C2, C3, C22, C6, C23, C9, C2×C6, C4○D4, C24, C18, C22×C6, C2×C4○D4, C2×C18, C3×C4○D4, C23×C6, C22×C18, C6×C4○D4, C9×C4○D4, C23×C18, C4○D4×C18

Smallest permutation representation of C4○D4×C18
On 144 points
Generators in S144
(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 60 132 123)(2 61 133 124)(3 62 134 125)(4 63 135 126)(5 64 136 109)(6 65 137 110)(7 66 138 111)(8 67 139 112)(9 68 140 113)(10 69 141 114)(11 70 142 115)(12 71 143 116)(13 72 144 117)(14 55 127 118)(15 56 128 119)(16 57 129 120)(17 58 130 121)(18 59 131 122)(19 74 50 107)(20 75 51 108)(21 76 52 91)(22 77 53 92)(23 78 54 93)(24 79 37 94)(25 80 38 95)(26 81 39 96)(27 82 40 97)(28 83 41 98)(29 84 42 99)(30 85 43 100)(31 86 44 101)(32 87 45 102)(33 88 46 103)(34 89 47 104)(35 90 48 105)(36 73 49 106)
(1 69 132 114)(2 70 133 115)(3 71 134 116)(4 72 135 117)(5 55 136 118)(6 56 137 119)(7 57 138 120)(8 58 139 121)(9 59 140 122)(10 60 141 123)(11 61 142 124)(12 62 143 125)(13 63 144 126)(14 64 127 109)(15 65 128 110)(16 66 129 111)(17 67 130 112)(18 68 131 113)(19 98 50 83)(20 99 51 84)(21 100 52 85)(22 101 53 86)(23 102 54 87)(24 103 37 88)(25 104 38 89)(26 105 39 90)(27 106 40 73)(28 107 41 74)(29 108 42 75)(30 91 43 76)(31 92 44 77)(32 93 45 78)(33 94 46 79)(34 95 47 80)(35 96 48 81)(36 97 49 82)
(1 33)(2 34)(3 35)(4 36)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(37 141)(38 142)(39 143)(40 144)(41 127)(42 128)(43 129)(44 130)(45 131)(46 132)(47 133)(48 134)(49 135)(50 136)(51 137)(52 138)(53 139)(54 140)(55 83)(56 84)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(71 81)(72 82)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 109)(108 110)

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

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

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

180 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 3A 3B 4A 4B 4C 4D 4E ··· 4J 6A ··· 6F 6G ··· 6R 9A ··· 9F 12A ··· 12H 12I ··· 12T 18A ··· 18R 18S ··· 18BB 36A ··· 36X 36Y ··· 36BH order 1 2 2 2 2 ··· 2 3 3 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 size 1 1 1 1 2 ··· 2 1 1 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

180 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 type + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 C9 C18 C18 C18 C18 C4○D4 C3×C4○D4 C9×C4○D4 kernel C4○D4×C18 C22×C36 D4×C18 Q8×C18 C9×C4○D4 C6×C4○D4 C22×C12 C6×D4 C6×Q8 C3×C4○D4 C2×C4○D4 C22×C4 C2×D4 C2×Q8 C4○D4 C18 C6 C2 # reps 1 3 3 1 8 2 6 6 2 16 6 18 18 6 48 4 8 24

Matrix representation of C4○D4×C18 in GL3(𝔽37) generated by

 36 0 0 0 21 0 0 0 21
,
 1 0 0 0 31 0 0 0 31
,
 1 0 0 0 6 0 0 13 31
,
 1 0 0 0 3 20 0 7 34
G:=sub<GL(3,GF(37))| [36,0,0,0,21,0,0,0,21],[1,0,0,0,31,0,0,0,31],[1,0,0,0,6,13,0,0,31],[1,0,0,0,3,7,0,20,34] >;

C4○D4×C18 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_{18}
% in TeX

G:=Group("C4oD4xC18");
// GroupNames label

G:=SmallGroup(288,370);
// by ID

G=gap.SmallGroup(288,370);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,701,268,242]);
// Polycyclic

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

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