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G = D7×C23order 322 = 2·7·23

Direct product of C23 and D7

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

Aliases: D7×C23, C7⋊C46, C1613C2, SmallGroup(322,1)

Series: Derived Chief Lower central Upper central

C1C7 — D7×C23
C1C7C161 — D7×C23
C7 — D7×C23
C1C23

Generators and relations for D7×C23
 G = < a,b,c | a23=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

7C2
7C46

Smallest permutation representation of D7×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)
(1 135 36 99 154 56 77)(2 136 37 100 155 57 78)(3 137 38 101 156 58 79)(4 138 39 102 157 59 80)(5 116 40 103 158 60 81)(6 117 41 104 159 61 82)(7 118 42 105 160 62 83)(8 119 43 106 161 63 84)(9 120 44 107 139 64 85)(10 121 45 108 140 65 86)(11 122 46 109 141 66 87)(12 123 24 110 142 67 88)(13 124 25 111 143 68 89)(14 125 26 112 144 69 90)(15 126 27 113 145 47 91)(16 127 28 114 146 48 92)(17 128 29 115 147 49 70)(18 129 30 93 148 50 71)(19 130 31 94 149 51 72)(20 131 32 95 150 52 73)(21 132 33 96 151 53 74)(22 133 34 97 152 54 75)(23 134 35 98 153 55 76)
(1 77)(2 78)(3 79)(4 80)(5 81)(6 82)(7 83)(8 84)(9 85)(10 86)(11 87)(12 88)(13 89)(14 90)(15 91)(16 92)(17 70)(18 71)(19 72)(20 73)(21 74)(22 75)(23 76)(24 142)(25 143)(26 144)(27 145)(28 146)(29 147)(30 148)(31 149)(32 150)(33 151)(34 152)(35 153)(36 154)(37 155)(38 156)(39 157)(40 158)(41 159)(42 160)(43 161)(44 139)(45 140)(46 141)(47 126)(48 127)(49 128)(50 129)(51 130)(52 131)(53 132)(54 133)(55 134)(56 135)(57 136)(58 137)(59 138)(60 116)(61 117)(62 118)(63 119)(64 120)(65 121)(66 122)(67 123)(68 124)(69 125)

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

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

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

115 conjugacy classes

class 1  2 7A7B7C23A···23V46A···46V161A···161BN
order1277723···2346···46161···161
size172221···17···72···2

115 irreducible representations

dim111122
type+++
imageC1C2C23C46D7D7×C23
kernelD7×C23C161D7C7C23C1
# reps112222366

Matrix representation of D7×C23 in GL2(𝔽967) generated by

6410
0641
,
2561
9660
,
01
10
G:=sub<GL(2,GF(967))| [641,0,0,641],[256,966,1,0],[0,1,1,0] >;

D7×C23 in GAP, Magma, Sage, TeX

D_7\times C_{23}
% in TeX

G:=Group("D7xC23");
// GroupNames label

G:=SmallGroup(322,1);
// by ID

G=gap.SmallGroup(322,1);
# by ID

G:=PCGroup([3,-2,-23,-7,2486]);
// Polycyclic

G:=Group<a,b,c|a^23=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×C23 in TeX

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