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

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

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

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

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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