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## G = D7×C18order 252 = 22·32·7

### Direct product of C18 and D7

Aliases: D7×C18, C1263C2, C143C18, C634C22, C42.11C6, C3.(C6×D7), C73(C2×C18), C6.3(C3×D7), C21.3(C2×C6), (C3×D7).2C6, (C6×D7).2C3, SmallGroup(252,12)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — D7×C18
 Chief series C1 — C7 — C21 — C63 — C9×D7 — D7×C18
 Lower central C7 — D7×C18
 Upper central C1 — C18

Generators and relations for D7×C18
G = < a,b,c | a18=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D7×C18
On 126 points
Generators in S126
(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)
(1 30 117 85 70 105 50)(2 31 118 86 71 106 51)(3 32 119 87 72 107 52)(4 33 120 88 55 108 53)(5 34 121 89 56 91 54)(6 35 122 90 57 92 37)(7 36 123 73 58 93 38)(8 19 124 74 59 94 39)(9 20 125 75 60 95 40)(10 21 126 76 61 96 41)(11 22 109 77 62 97 42)(12 23 110 78 63 98 43)(13 24 111 79 64 99 44)(14 25 112 80 65 100 45)(15 26 113 81 66 101 46)(16 27 114 82 67 102 47)(17 28 115 83 68 103 48)(18 29 116 84 69 104 49)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 37)(16 38)(17 39)(18 40)(19 103)(20 104)(21 105)(22 106)(23 107)(24 108)(25 91)(26 92)(27 93)(28 94)(29 95)(30 96)(31 97)(32 98)(33 99)(34 100)(35 101)(36 102)(55 111)(56 112)(57 113)(58 114)(59 115)(60 116)(61 117)(62 118)(63 119)(64 120)(65 121)(66 122)(67 123)(68 124)(69 125)(70 126)(71 109)(72 110)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 3A 3B 6A 6B 6C 6D 6E 6F 7A 7B 7C 9A ··· 9F 14A 14B 14C 18A ··· 18F 18G ··· 18R 21A ··· 21F 42A ··· 42F 63A ··· 63R 126A ··· 126R order 1 2 2 2 3 3 6 6 6 6 6 6 7 7 7 9 ··· 9 14 14 14 18 ··· 18 18 ··· 18 21 ··· 21 42 ··· 42 63 ··· 63 126 ··· 126 size 1 1 7 7 1 1 1 1 7 7 7 7 2 2 2 1 ··· 1 2 2 2 1 ··· 1 7 ··· 7 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + image C1 C2 C2 C3 C6 C6 C9 C18 C18 D7 D14 C3×D7 C6×D7 C9×D7 D7×C18 kernel D7×C18 C9×D7 C126 C6×D7 C3×D7 C42 D14 D7 C14 C18 C9 C6 C3 C2 C1 # reps 1 2 1 2 4 2 6 12 6 3 3 6 6 18 18

Matrix representation of D7×C18 in GL3(𝔽127) generated by

 24 0 0 0 107 0 0 0 107
,
 1 0 0 0 0 1 0 126 36
,
 1 0 0 0 0 1 0 1 0
G:=sub<GL(3,GF(127))| [24,0,0,0,107,0,0,0,107],[1,0,0,0,0,126,0,1,36],[1,0,0,0,0,1,0,1,0] >;

D7×C18 in GAP, Magma, Sage, TeX

D_7\times C_{18}
% in TeX

G:=Group("D7xC18");
// GroupNames label

G:=SmallGroup(252,12);
// by ID

G=gap.SmallGroup(252,12);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-7,57,5404]);
// Polycyclic

G:=Group<a,b,c|a^18=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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