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G = C7×D23order 322 = 2·7·23

Direct product of C7 and D23

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

Aliases: C7×D23, C23⋊C14, C1612C2, SmallGroup(322,2)

Series: Derived Chief Lower central Upper central

C1C23 — C7×D23
C1C23C161 — C7×D23
C23 — C7×D23
C1C7

Generators and relations for C7×D23
 G = < a,b,c | a7=b23=c2=1, ab=ba, ac=ca, cbc=b-1 >

23C2
23C14

Smallest permutation representation of C7×D23
On 161 points
Generators in S161
(1 139 131 112 87 48 34)(2 140 132 113 88 49 35)(3 141 133 114 89 50 36)(4 142 134 115 90 51 37)(5 143 135 93 91 52 38)(6 144 136 94 92 53 39)(7 145 137 95 70 54 40)(8 146 138 96 71 55 41)(9 147 116 97 72 56 42)(10 148 117 98 73 57 43)(11 149 118 99 74 58 44)(12 150 119 100 75 59 45)(13 151 120 101 76 60 46)(14 152 121 102 77 61 24)(15 153 122 103 78 62 25)(16 154 123 104 79 63 26)(17 155 124 105 80 64 27)(18 156 125 106 81 65 28)(19 157 126 107 82 66 29)(20 158 127 108 83 67 30)(21 159 128 109 84 68 31)(22 160 129 110 85 69 32)(23 161 130 111 86 47 33)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(24 43)(25 42)(26 41)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(44 46)(47 48)(49 69)(50 68)(51 67)(52 66)(53 65)(54 64)(55 63)(56 62)(57 61)(58 60)(70 80)(71 79)(72 78)(73 77)(74 76)(81 92)(82 91)(83 90)(84 89)(85 88)(86 87)(93 107)(94 106)(95 105)(96 104)(97 103)(98 102)(99 101)(108 115)(109 114)(110 113)(111 112)(116 122)(117 121)(118 120)(123 138)(124 137)(125 136)(126 135)(127 134)(128 133)(129 132)(130 131)(139 161)(140 160)(141 159)(142 158)(143 157)(144 156)(145 155)(146 154)(147 153)(148 152)(149 151)

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

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

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

91 conjugacy classes

class 1  2 7A···7F14A···14F23A···23K161A···161BN
order127···714···1423···23161···161
size1231···123···232···22···2

91 irreducible representations

dim111122
type+++
imageC1C2C7C14D23C7×D23
kernelC7×D23C161D23C23C7C1
# reps11661166

Matrix representation of C7×D23 in GL2(𝔽967) generated by

970
097
,
535313
966914
,
660226
182307
G:=sub<GL(2,GF(967))| [97,0,0,97],[535,966,313,914],[660,182,226,307] >;

C7×D23 in GAP, Magma, Sage, TeX

C_7\times D_{23}
% in TeX

G:=Group("C7xD23");
// GroupNames label

G:=SmallGroup(322,2);
// by ID

G=gap.SmallGroup(322,2);
# by ID

G:=PCGroup([3,-2,-7,-23,2774]);
// Polycyclic

G:=Group<a,b,c|a^7=b^23=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C7×D23 in TeX

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