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## G = C19×C7⋊C3order 399 = 3·7·19

### Direct product of C19 and C7⋊C3

Aliases: C19×C7⋊C3, C7⋊C57, C1331C3, SmallGroup(399,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C19×C7⋊C3
 Chief series C1 — C7 — C133 — C19×C7⋊C3
 Lower central C7 — C19×C7⋊C3
 Upper central C1 — C19

Generators and relations for C19×C7⋊C3
G = < a,b,c | a19=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C19×C7⋊C3
On 133 points
Generators in S133
(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)
(1 20 65 110 90 46 130)(2 21 66 111 91 47 131)(3 22 67 112 92 48 132)(4 23 68 113 93 49 133)(5 24 69 114 94 50 115)(6 25 70 96 95 51 116)(7 26 71 97 77 52 117)(8 27 72 98 78 53 118)(9 28 73 99 79 54 119)(10 29 74 100 80 55 120)(11 30 75 101 81 56 121)(12 31 76 102 82 57 122)(13 32 58 103 83 39 123)(14 33 59 104 84 40 124)(15 34 60 105 85 41 125)(16 35 61 106 86 42 126)(17 36 62 107 87 43 127)(18 37 63 108 88 44 128)(19 38 64 109 89 45 129)
(20 65 90)(21 66 91)(22 67 92)(23 68 93)(24 69 94)(25 70 95)(26 71 77)(27 72 78)(28 73 79)(29 74 80)(30 75 81)(31 76 82)(32 58 83)(33 59 84)(34 60 85)(35 61 86)(36 62 87)(37 63 88)(38 64 89)(39 103 123)(40 104 124)(41 105 125)(42 106 126)(43 107 127)(44 108 128)(45 109 129)(46 110 130)(47 111 131)(48 112 132)(49 113 133)(50 114 115)(51 96 116)(52 97 117)(53 98 118)(54 99 119)(55 100 120)(56 101 121)(57 102 122)

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

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

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

95 conjugacy classes

 class 1 3A 3B 7A 7B 19A ··· 19R 57A ··· 57AJ 133A ··· 133AJ order 1 3 3 7 7 19 ··· 19 57 ··· 57 133 ··· 133 size 1 7 7 3 3 1 ··· 1 7 ··· 7 3 ··· 3

95 irreducible representations

 dim 1 1 1 1 3 3 type + image C1 C3 C19 C57 C7⋊C3 C19×C7⋊C3 kernel C19×C7⋊C3 C133 C7⋊C3 C7 C19 C1 # reps 1 2 18 36 2 36

Matrix representation of C19×C7⋊C3 in GL3(𝔽1597) generated by

 81 0 0 0 81 0 0 0 81
,
 57 1 1540 1 0 1596 0 1 1596
,
 1596 0 1 56 1 1541 1596 0 0
G:=sub<GL(3,GF(1597))| [81,0,0,0,81,0,0,0,81],[57,1,0,1,0,1,1540,1596,1596],[1596,56,1596,0,1,0,1,1541,0] >;

C19×C7⋊C3 in GAP, Magma, Sage, TeX

C_{19}\times C_7\rtimes C_3
% in TeX

G:=Group("C19xC7:C3");
// GroupNames label

G:=SmallGroup(399,1);
// by ID

G=gap.SmallGroup(399,1);
# by ID

G:=PCGroup([3,-3,-19,-7,1028]);
// Polycyclic

G:=Group<a,b,c|a^19=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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