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## G = C7×D19order 266 = 2·7·19

### Direct product of C7 and D19

Aliases: C7×D19, C19⋊C14, C1332C2, SmallGroup(266,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C19 — C7×D19
 Chief series C1 — C19 — C133 — C7×D19
 Lower central C19 — C7×D19
 Upper central C1 — C7

Generators and relations for C7×D19
G = < a,b,c | a7=b19=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C7×D19
On 133 points
Generators in S133
(1 116 108 94 71 46 33)(2 117 109 95 72 47 34)(3 118 110 77 73 48 35)(4 119 111 78 74 49 36)(5 120 112 79 75 50 37)(6 121 113 80 76 51 38)(7 122 114 81 58 52 20)(8 123 96 82 59 53 21)(9 124 97 83 60 54 22)(10 125 98 84 61 55 23)(11 126 99 85 62 56 24)(12 127 100 86 63 57 25)(13 128 101 87 64 39 26)(14 129 102 88 65 40 27)(15 130 103 89 66 41 28)(16 131 104 90 67 42 29)(17 132 105 91 68 43 30)(18 133 106 92 69 44 31)(19 115 107 93 70 45 32)
(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 19)(2 18)(3 17)(4 16)(5 15)(6 14)(7 13)(8 12)(9 11)(20 26)(21 25)(22 24)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(53 57)(54 56)(58 64)(59 63)(60 62)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(77 91)(78 90)(79 89)(80 88)(81 87)(82 86)(83 85)(92 95)(93 94)(96 100)(97 99)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(115 116)(117 133)(118 132)(119 131)(120 130)(121 129)(122 128)(123 127)(124 126)

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

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

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

77 conjugacy classes

 class 1 2 7A ··· 7F 14A ··· 14F 19A ··· 19I 133A ··· 133BB order 1 2 7 ··· 7 14 ··· 14 19 ··· 19 133 ··· 133 size 1 19 1 ··· 1 19 ··· 19 2 ··· 2 2 ··· 2

77 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C7 C14 D19 C7×D19 kernel C7×D19 C133 D19 C19 C7 C1 # reps 1 1 6 6 9 54

Matrix representation of C7×D19 in GL2(𝔽1597) generated by

 1150 0 0 1150
,
 159 1 1372 320
,
 185 551 448 1412
G:=sub<GL(2,GF(1597))| [1150,0,0,1150],[159,1372,1,320],[185,448,551,1412] >;

C7×D19 in GAP, Magma, Sage, TeX

C_7\times D_{19}
% in TeX

G:=Group("C7xD19");
// GroupNames label

G:=SmallGroup(266,2);
// by ID

G=gap.SmallGroup(266,2);
# by ID

G:=PCGroup([3,-2,-7,-19,2270]);
// Polycyclic

G:=Group<a,b,c|a^7=b^19=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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