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G = D7×C18order 252 = 22·32·7

Direct product of C18 and D7

direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary

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

C1C7 — D7×C18
C1C7C21C63C9×D7 — D7×C18
C7 — D7×C18
C1C18

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

7C2
7C2
7C22
7C6
7C6
7C2×C6
7C18
7C18
7C2×C18

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 2A2B2C3A3B6A6B6C6D6E6F7A7B7C9A···9F14A14B14C18A···18F18G···18R21A···21F42A···42F63A···63R126A···126R
order1222336666667779···914141418···1818···1821···2142···4263···63126···126
size1177111177772221···12221···17···72···22···22···22···2

90 irreducible representations

dim111111111222222
type+++++
imageC1C2C2C3C6C6C9C18C18D7D14C3×D7C6×D7C9×D7D7×C18
kernelD7×C18C9×D7C126C6×D7C3×D7C42D14D7C14C18C9C6C3C2C1
# reps121242612633661818

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

2400
01070
00107
,
100
001
012636
,
100
001
010
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

Export

Subgroup lattice of D7×C18 in TeX

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