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## G = C18×C7⋊C3order 378 = 2·33·7

### Direct product of C18 and C7⋊C3

Aliases: C18×C7⋊C3, C6313C6, C1261C3, C42.1C32, C7⋊C99C6, C141(C3×C9), C73(C3×C18), C21.7(C3×C6), (C2×C7⋊C9)⋊4C3, C6.1(C3×C7⋊C3), C3.1(C6×C7⋊C3), (C3×C7⋊C3).6C6, (C6×C7⋊C3).3C3, SmallGroup(378,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C18×C7⋊C3
 Chief series C1 — C7 — C21 — C63 — C9×C7⋊C3 — C18×C7⋊C3
 Lower central C7 — C18×C7⋊C3
 Upper central C1 — C18

Generators and relations for C18×C7⋊C3
G = < a,b,c | a18=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C18×C7⋊C3
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)
(1 126 53 25 83 99 69)(2 109 54 26 84 100 70)(3 110 37 27 85 101 71)(4 111 38 28 86 102 72)(5 112 39 29 87 103 55)(6 113 40 30 88 104 56)(7 114 41 31 89 105 57)(8 115 42 32 90 106 58)(9 116 43 33 73 107 59)(10 117 44 34 74 108 60)(11 118 45 35 75 91 61)(12 119 46 36 76 92 62)(13 120 47 19 77 93 63)(14 121 48 20 78 94 64)(15 122 49 21 79 95 65)(16 123 50 22 80 96 66)(17 124 51 23 81 97 67)(18 125 52 24 82 98 68)
(19 63 93)(20 64 94)(21 65 95)(22 66 96)(23 67 97)(24 68 98)(25 69 99)(26 70 100)(27 71 101)(28 72 102)(29 55 103)(30 56 104)(31 57 105)(32 58 106)(33 59 107)(34 60 108)(35 61 91)(36 62 92)(37 85 110)(38 86 111)(39 87 112)(40 88 113)(41 89 114)(42 90 115)(43 73 116)(44 74 117)(45 75 118)(46 76 119)(47 77 120)(48 78 121)(49 79 122)(50 80 123)(51 81 124)(52 82 125)(53 83 126)(54 84 109)

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

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

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

90 conjugacy classes

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

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 type + + image C1 C2 C3 C3 C3 C6 C6 C6 C9 C18 C7⋊C3 C2×C7⋊C3 C3×C7⋊C3 C6×C7⋊C3 C9×C7⋊C3 C18×C7⋊C3 kernel C18×C7⋊C3 C9×C7⋊C3 C2×C7⋊C9 C126 C6×C7⋊C3 C7⋊C9 C63 C3×C7⋊C3 C2×C7⋊C3 C7⋊C3 C18 C9 C6 C3 C2 C1 # reps 1 1 4 2 2 4 2 2 18 18 2 2 4 4 12 12

Matrix representation of C18×C7⋊C3 in GL3(𝔽127) generated by

 105 0 0 0 105 0 0 0 105
,
 126 126 104 1 0 1 0 1 23
,
 0 105 1 1 1 23 0 23 126
G:=sub<GL(3,GF(127))| [105,0,0,0,105,0,0,0,105],[126,1,0,126,0,1,104,1,23],[0,1,0,105,1,23,1,23,126] >;

C18×C7⋊C3 in GAP, Magma, Sage, TeX

C_{18}\times C_7\rtimes C_3
% in TeX

G:=Group("C18xC7:C3");
// GroupNames label

G:=SmallGroup(378,23);
// by ID

G=gap.SmallGroup(378,23);
# by ID

G:=PCGroup([5,-2,-3,-3,-3,-7,57,1359]);
// Polycyclic

G:=Group<a,b,c|a^18=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations

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