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## G = C7⋊C3×C23order 483 = 3·7·23

### Direct product of C23 and C7⋊C3

Aliases: C7⋊C3×C23, C7⋊C69, C161⋊C3, SmallGroup(483,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C7⋊C3×C23
 Chief series C1 — C7 — C161 — C7⋊C3×C23
 Lower central C7 — C7⋊C3×C23
 Upper central C1 — C23

Generators and relations for C7⋊C3×C23
G = < a,b,c | a23=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >

Smallest permutation representation of C7⋊C3×C23
On 161 points
Generators in S161
(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)(116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138)(139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161)
(1 27 108 65 156 80 122)(2 28 109 66 157 81 123)(3 29 110 67 158 82 124)(4 30 111 68 159 83 125)(5 31 112 69 160 84 126)(6 32 113 47 161 85 127)(7 33 114 48 139 86 128)(8 34 115 49 140 87 129)(9 35 93 50 141 88 130)(10 36 94 51 142 89 131)(11 37 95 52 143 90 132)(12 38 96 53 144 91 133)(13 39 97 54 145 92 134)(14 40 98 55 146 70 135)(15 41 99 56 147 71 136)(16 42 100 57 148 72 137)(17 43 101 58 149 73 138)(18 44 102 59 150 74 116)(19 45 103 60 151 75 117)(20 46 104 61 152 76 118)(21 24 105 62 153 77 119)(22 25 106 63 154 78 120)(23 26 107 64 155 79 121)
(24 105 153)(25 106 154)(26 107 155)(27 108 156)(28 109 157)(29 110 158)(30 111 159)(31 112 160)(32 113 161)(33 114 139)(34 115 140)(35 93 141)(36 94 142)(37 95 143)(38 96 144)(39 97 145)(40 98 146)(41 99 147)(42 100 148)(43 101 149)(44 102 150)(45 103 151)(46 104 152)(47 127 85)(48 128 86)(49 129 87)(50 130 88)(51 131 89)(52 132 90)(53 133 91)(54 134 92)(55 135 70)(56 136 71)(57 137 72)(58 138 73)(59 116 74)(60 117 75)(61 118 76)(62 119 77)(63 120 78)(64 121 79)(65 122 80)(66 123 81)(67 124 82)(68 125 83)(69 126 84)

G:=sub<Sym(161)| (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)(116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138)(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161), (1,27,108,65,156,80,122)(2,28,109,66,157,81,123)(3,29,110,67,158,82,124)(4,30,111,68,159,83,125)(5,31,112,69,160,84,126)(6,32,113,47,161,85,127)(7,33,114,48,139,86,128)(8,34,115,49,140,87,129)(9,35,93,50,141,88,130)(10,36,94,51,142,89,131)(11,37,95,52,143,90,132)(12,38,96,53,144,91,133)(13,39,97,54,145,92,134)(14,40,98,55,146,70,135)(15,41,99,56,147,71,136)(16,42,100,57,148,72,137)(17,43,101,58,149,73,138)(18,44,102,59,150,74,116)(19,45,103,60,151,75,117)(20,46,104,61,152,76,118)(21,24,105,62,153,77,119)(22,25,106,63,154,78,120)(23,26,107,64,155,79,121), (24,105,153)(25,106,154)(26,107,155)(27,108,156)(28,109,157)(29,110,158)(30,111,159)(31,112,160)(32,113,161)(33,114,139)(34,115,140)(35,93,141)(36,94,142)(37,95,143)(38,96,144)(39,97,145)(40,98,146)(41,99,147)(42,100,148)(43,101,149)(44,102,150)(45,103,151)(46,104,152)(47,127,85)(48,128,86)(49,129,87)(50,130,88)(51,131,89)(52,132,90)(53,133,91)(54,134,92)(55,135,70)(56,136,71)(57,137,72)(58,138,73)(59,116,74)(60,117,75)(61,118,76)(62,119,77)(63,120,78)(64,121,79)(65,122,80)(66,123,81)(67,124,82)(68,125,83)(69,126,84)>;

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)(116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138)(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161), (1,27,108,65,156,80,122)(2,28,109,66,157,81,123)(3,29,110,67,158,82,124)(4,30,111,68,159,83,125)(5,31,112,69,160,84,126)(6,32,113,47,161,85,127)(7,33,114,48,139,86,128)(8,34,115,49,140,87,129)(9,35,93,50,141,88,130)(10,36,94,51,142,89,131)(11,37,95,52,143,90,132)(12,38,96,53,144,91,133)(13,39,97,54,145,92,134)(14,40,98,55,146,70,135)(15,41,99,56,147,71,136)(16,42,100,57,148,72,137)(17,43,101,58,149,73,138)(18,44,102,59,150,74,116)(19,45,103,60,151,75,117)(20,46,104,61,152,76,118)(21,24,105,62,153,77,119)(22,25,106,63,154,78,120)(23,26,107,64,155,79,121), (24,105,153)(25,106,154)(26,107,155)(27,108,156)(28,109,157)(29,110,158)(30,111,159)(31,112,160)(32,113,161)(33,114,139)(34,115,140)(35,93,141)(36,94,142)(37,95,143)(38,96,144)(39,97,145)(40,98,146)(41,99,147)(42,100,148)(43,101,149)(44,102,150)(45,103,151)(46,104,152)(47,127,85)(48,128,86)(49,129,87)(50,130,88)(51,131,89)(52,132,90)(53,133,91)(54,134,92)(55,135,70)(56,136,71)(57,137,72)(58,138,73)(59,116,74)(60,117,75)(61,118,76)(62,119,77)(63,120,78)(64,121,79)(65,122,80)(66,123,81)(67,124,82)(68,125,83)(69,126,84) );

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),(116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138),(139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161)], [(1,27,108,65,156,80,122),(2,28,109,66,157,81,123),(3,29,110,67,158,82,124),(4,30,111,68,159,83,125),(5,31,112,69,160,84,126),(6,32,113,47,161,85,127),(7,33,114,48,139,86,128),(8,34,115,49,140,87,129),(9,35,93,50,141,88,130),(10,36,94,51,142,89,131),(11,37,95,52,143,90,132),(12,38,96,53,144,91,133),(13,39,97,54,145,92,134),(14,40,98,55,146,70,135),(15,41,99,56,147,71,136),(16,42,100,57,148,72,137),(17,43,101,58,149,73,138),(18,44,102,59,150,74,116),(19,45,103,60,151,75,117),(20,46,104,61,152,76,118),(21,24,105,62,153,77,119),(22,25,106,63,154,78,120),(23,26,107,64,155,79,121)], [(24,105,153),(25,106,154),(26,107,155),(27,108,156),(28,109,157),(29,110,158),(30,111,159),(31,112,160),(32,113,161),(33,114,139),(34,115,140),(35,93,141),(36,94,142),(37,95,143),(38,96,144),(39,97,145),(40,98,146),(41,99,147),(42,100,148),(43,101,149),(44,102,150),(45,103,151),(46,104,152),(47,127,85),(48,128,86),(49,129,87),(50,130,88),(51,131,89),(52,132,90),(53,133,91),(54,134,92),(55,135,70),(56,136,71),(57,137,72),(58,138,73),(59,116,74),(60,117,75),(61,118,76),(62,119,77),(63,120,78),(64,121,79),(65,122,80),(66,123,81),(67,124,82),(68,125,83),(69,126,84)])

115 conjugacy classes

 class 1 3A 3B 7A 7B 23A ··· 23V 69A ··· 69AR 161A ··· 161AR order 1 3 3 7 7 23 ··· 23 69 ··· 69 161 ··· 161 size 1 7 7 3 3 1 ··· 1 7 ··· 7 3 ··· 3

115 irreducible representations

 dim 1 1 1 1 3 3 type + image C1 C3 C23 C69 C7⋊C3 C7⋊C3×C23 kernel C7⋊C3×C23 C161 C7⋊C3 C7 C23 C1 # reps 1 2 22 44 2 44

Matrix representation of C7⋊C3×C23 in GL3(𝔽967) generated by

 69 0 0 0 69 0 0 0 69
,
 966 966 699 1 0 1 0 1 268
,
 0 700 1 1 1 268 0 268 966
G:=sub<GL(3,GF(967))| [69,0,0,0,69,0,0,0,69],[966,1,0,966,0,1,699,1,268],[0,1,0,700,1,268,1,268,966] >;

C7⋊C3×C23 in GAP, Magma, Sage, TeX

C_7\rtimes C_3\times C_{23}
% in TeX

G:=Group("C7:C3xC23");
// GroupNames label

G:=SmallGroup(483,1);
// by ID

G=gap.SmallGroup(483,1);
# by ID

G:=PCGroup([3,-3,-23,-7,1244]);
// Polycyclic

G:=Group<a,b,c|a^23=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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