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G = C10×D23order 460 = 22·5·23

Direct product of C10 and D23

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

Aliases: C10×D23, C46⋊C10, C2302C2, C1153C22, C23⋊(C2×C10), SmallGroup(460,8)

Series: Derived Chief Lower central Upper central

C1C23 — C10×D23
C1C23C115C5×D23 — C10×D23
C23 — C10×D23
C1C10

Generators and relations for C10×D23
 G = < a,b,c | a10=b23=c2=1, ab=ba, ac=ca, cbc=b-1 >

23C2
23C2
23C22
23C10
23C10
23C2×C10

Smallest permutation representation of C10×D23
On 230 points
Generators in S230
(1 183 114 146 88 130 53 209 37 193)(2 184 115 147 89 131 54 210 38 194)(3 162 93 148 90 132 55 211 39 195)(4 163 94 149 91 133 56 212 40 196)(5 164 95 150 92 134 57 213 41 197)(6 165 96 151 70 135 58 214 42 198)(7 166 97 152 71 136 59 215 43 199)(8 167 98 153 72 137 60 216 44 200)(9 168 99 154 73 138 61 217 45 201)(10 169 100 155 74 116 62 218 46 202)(11 170 101 156 75 117 63 219 24 203)(12 171 102 157 76 118 64 220 25 204)(13 172 103 158 77 119 65 221 26 205)(14 173 104 159 78 120 66 222 27 206)(15 174 105 160 79 121 67 223 28 207)(16 175 106 161 80 122 68 224 29 185)(17 176 107 139 81 123 69 225 30 186)(18 177 108 140 82 124 47 226 31 187)(19 178 109 141 83 125 48 227 32 188)(20 179 110 142 84 126 49 228 33 189)(21 180 111 143 85 127 50 229 34 190)(22 181 112 144 86 128 51 230 35 191)(23 182 113 145 87 129 52 208 36 192)
(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)(162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184)(185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207)(208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230)
(1 129)(2 128)(3 127)(4 126)(5 125)(6 124)(7 123)(8 122)(9 121)(10 120)(11 119)(12 118)(13 117)(14 116)(15 138)(16 137)(17 136)(18 135)(19 134)(20 133)(21 132)(22 131)(23 130)(24 158)(25 157)(26 156)(27 155)(28 154)(29 153)(30 152)(31 151)(32 150)(33 149)(34 148)(35 147)(36 146)(37 145)(38 144)(39 143)(40 142)(41 141)(42 140)(43 139)(44 161)(45 160)(46 159)(47 165)(48 164)(49 163)(50 162)(51 184)(52 183)(53 182)(54 181)(55 180)(56 179)(57 178)(58 177)(59 176)(60 175)(61 174)(62 173)(63 172)(64 171)(65 170)(66 169)(67 168)(68 167)(69 166)(70 187)(71 186)(72 185)(73 207)(74 206)(75 205)(76 204)(77 203)(78 202)(79 201)(80 200)(81 199)(82 198)(83 197)(84 196)(85 195)(86 194)(87 193)(88 192)(89 191)(90 190)(91 189)(92 188)(93 229)(94 228)(95 227)(96 226)(97 225)(98 224)(99 223)(100 222)(101 221)(102 220)(103 219)(104 218)(105 217)(106 216)(107 215)(108 214)(109 213)(110 212)(111 211)(112 210)(113 209)(114 208)(115 230)

G:=sub<Sym(230)| (1,183,114,146,88,130,53,209,37,193)(2,184,115,147,89,131,54,210,38,194)(3,162,93,148,90,132,55,211,39,195)(4,163,94,149,91,133,56,212,40,196)(5,164,95,150,92,134,57,213,41,197)(6,165,96,151,70,135,58,214,42,198)(7,166,97,152,71,136,59,215,43,199)(8,167,98,153,72,137,60,216,44,200)(9,168,99,154,73,138,61,217,45,201)(10,169,100,155,74,116,62,218,46,202)(11,170,101,156,75,117,63,219,24,203)(12,171,102,157,76,118,64,220,25,204)(13,172,103,158,77,119,65,221,26,205)(14,173,104,159,78,120,66,222,27,206)(15,174,105,160,79,121,67,223,28,207)(16,175,106,161,80,122,68,224,29,185)(17,176,107,139,81,123,69,225,30,186)(18,177,108,140,82,124,47,226,31,187)(19,178,109,141,83,125,48,227,32,188)(20,179,110,142,84,126,49,228,33,189)(21,180,111,143,85,127,50,229,34,190)(22,181,112,144,86,128,51,230,35,191)(23,182,113,145,87,129,52,208,36,192), (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)(162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207)(208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230), (1,129)(2,128)(3,127)(4,126)(5,125)(6,124)(7,123)(8,122)(9,121)(10,120)(11,119)(12,118)(13,117)(14,116)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,132)(22,131)(23,130)(24,158)(25,157)(26,156)(27,155)(28,154)(29,153)(30,152)(31,151)(32,150)(33,149)(34,148)(35,147)(36,146)(37,145)(38,144)(39,143)(40,142)(41,141)(42,140)(43,139)(44,161)(45,160)(46,159)(47,165)(48,164)(49,163)(50,162)(51,184)(52,183)(53,182)(54,181)(55,180)(56,179)(57,178)(58,177)(59,176)(60,175)(61,174)(62,173)(63,172)(64,171)(65,170)(66,169)(67,168)(68,167)(69,166)(70,187)(71,186)(72,185)(73,207)(74,206)(75,205)(76,204)(77,203)(78,202)(79,201)(80,200)(81,199)(82,198)(83,197)(84,196)(85,195)(86,194)(87,193)(88,192)(89,191)(90,190)(91,189)(92,188)(93,229)(94,228)(95,227)(96,226)(97,225)(98,224)(99,223)(100,222)(101,221)(102,220)(103,219)(104,218)(105,217)(106,216)(107,215)(108,214)(109,213)(110,212)(111,211)(112,210)(113,209)(114,208)(115,230)>;

G:=Group( (1,183,114,146,88,130,53,209,37,193)(2,184,115,147,89,131,54,210,38,194)(3,162,93,148,90,132,55,211,39,195)(4,163,94,149,91,133,56,212,40,196)(5,164,95,150,92,134,57,213,41,197)(6,165,96,151,70,135,58,214,42,198)(7,166,97,152,71,136,59,215,43,199)(8,167,98,153,72,137,60,216,44,200)(9,168,99,154,73,138,61,217,45,201)(10,169,100,155,74,116,62,218,46,202)(11,170,101,156,75,117,63,219,24,203)(12,171,102,157,76,118,64,220,25,204)(13,172,103,158,77,119,65,221,26,205)(14,173,104,159,78,120,66,222,27,206)(15,174,105,160,79,121,67,223,28,207)(16,175,106,161,80,122,68,224,29,185)(17,176,107,139,81,123,69,225,30,186)(18,177,108,140,82,124,47,226,31,187)(19,178,109,141,83,125,48,227,32,188)(20,179,110,142,84,126,49,228,33,189)(21,180,111,143,85,127,50,229,34,190)(22,181,112,144,86,128,51,230,35,191)(23,182,113,145,87,129,52,208,36,192), (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)(162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207)(208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230), (1,129)(2,128)(3,127)(4,126)(5,125)(6,124)(7,123)(8,122)(9,121)(10,120)(11,119)(12,118)(13,117)(14,116)(15,138)(16,137)(17,136)(18,135)(19,134)(20,133)(21,132)(22,131)(23,130)(24,158)(25,157)(26,156)(27,155)(28,154)(29,153)(30,152)(31,151)(32,150)(33,149)(34,148)(35,147)(36,146)(37,145)(38,144)(39,143)(40,142)(41,141)(42,140)(43,139)(44,161)(45,160)(46,159)(47,165)(48,164)(49,163)(50,162)(51,184)(52,183)(53,182)(54,181)(55,180)(56,179)(57,178)(58,177)(59,176)(60,175)(61,174)(62,173)(63,172)(64,171)(65,170)(66,169)(67,168)(68,167)(69,166)(70,187)(71,186)(72,185)(73,207)(74,206)(75,205)(76,204)(77,203)(78,202)(79,201)(80,200)(81,199)(82,198)(83,197)(84,196)(85,195)(86,194)(87,193)(88,192)(89,191)(90,190)(91,189)(92,188)(93,229)(94,228)(95,227)(96,226)(97,225)(98,224)(99,223)(100,222)(101,221)(102,220)(103,219)(104,218)(105,217)(106,216)(107,215)(108,214)(109,213)(110,212)(111,211)(112,210)(113,209)(114,208)(115,230) );

G=PermutationGroup([(1,183,114,146,88,130,53,209,37,193),(2,184,115,147,89,131,54,210,38,194),(3,162,93,148,90,132,55,211,39,195),(4,163,94,149,91,133,56,212,40,196),(5,164,95,150,92,134,57,213,41,197),(6,165,96,151,70,135,58,214,42,198),(7,166,97,152,71,136,59,215,43,199),(8,167,98,153,72,137,60,216,44,200),(9,168,99,154,73,138,61,217,45,201),(10,169,100,155,74,116,62,218,46,202),(11,170,101,156,75,117,63,219,24,203),(12,171,102,157,76,118,64,220,25,204),(13,172,103,158,77,119,65,221,26,205),(14,173,104,159,78,120,66,222,27,206),(15,174,105,160,79,121,67,223,28,207),(16,175,106,161,80,122,68,224,29,185),(17,176,107,139,81,123,69,225,30,186),(18,177,108,140,82,124,47,226,31,187),(19,178,109,141,83,125,48,227,32,188),(20,179,110,142,84,126,49,228,33,189),(21,180,111,143,85,127,50,229,34,190),(22,181,112,144,86,128,51,230,35,191),(23,182,113,145,87,129,52,208,36,192)], [(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),(162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184),(185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207),(208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230)], [(1,129),(2,128),(3,127),(4,126),(5,125),(6,124),(7,123),(8,122),(9,121),(10,120),(11,119),(12,118),(13,117),(14,116),(15,138),(16,137),(17,136),(18,135),(19,134),(20,133),(21,132),(22,131),(23,130),(24,158),(25,157),(26,156),(27,155),(28,154),(29,153),(30,152),(31,151),(32,150),(33,149),(34,148),(35,147),(36,146),(37,145),(38,144),(39,143),(40,142),(41,141),(42,140),(43,139),(44,161),(45,160),(46,159),(47,165),(48,164),(49,163),(50,162),(51,184),(52,183),(53,182),(54,181),(55,180),(56,179),(57,178),(58,177),(59,176),(60,175),(61,174),(62,173),(63,172),(64,171),(65,170),(66,169),(67,168),(68,167),(69,166),(70,187),(71,186),(72,185),(73,207),(74,206),(75,205),(76,204),(77,203),(78,202),(79,201),(80,200),(81,199),(82,198),(83,197),(84,196),(85,195),(86,194),(87,193),(88,192),(89,191),(90,190),(91,189),(92,188),(93,229),(94,228),(95,227),(96,226),(97,225),(98,224),(99,223),(100,222),(101,221),(102,220),(103,219),(104,218),(105,217),(106,216),(107,215),(108,214),(109,213),(110,212),(111,211),(112,210),(113,209),(114,208),(115,230)])

130 conjugacy classes

class 1 2A2B2C5A5B5C5D10A10B10C10D10E···10L23A···23K46A···46K115A···115AR230A···230AR
order122255551010101010···1023···2346···46115···115230···230
size1123231111111123···232···22···22···22···2

130 irreducible representations

dim1111112222
type+++++
imageC1C2C2C5C10C10D23D46C5×D23C10×D23
kernelC10×D23C5×D23C230D46D23C46C10C5C2C1
# reps12148411114444

Matrix representation of C10×D23 in GL3(𝔽461) generated by

34700
04600
00460
,
100
04541
0175370
,
100
0911
018370
G:=sub<GL(3,GF(461))| [347,0,0,0,460,0,0,0,460],[1,0,0,0,454,175,0,1,370],[1,0,0,0,91,18,0,1,370] >;

C10×D23 in GAP, Magma, Sage, TeX

C_{10}\times D_{23}
% in TeX

G:=Group("C10xD23");
// GroupNames label

G:=SmallGroup(460,8);
// by ID

G=gap.SmallGroup(460,8);
# by ID

G:=PCGroup([4,-2,-2,-5,-23,7043]);
// Polycyclic

G:=Group<a,b,c|a^10=b^23=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C10×D23 in TeX

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