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## G = C10×D24order 480 = 25·3·5

### Direct product of C10 and D24

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C10×D24
 Chief series C1 — C3 — C6 — C12 — C60 — C5×D12 — C10×D12 — C10×D24
 Lower central C3 — C6 — C12 — C10×D24
 Upper central C1 — C2×C10 — C2×C20 — C2×C40

Generators and relations for C10×D24
G = < a,b,c | a10=b24=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 516 in 152 conjugacy classes, 66 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, D4, C23, C10, C10, C10, C12, D6, C2×C6, C15, C2×C8, D8, C2×D4, C20, C2×C10, C2×C10, C24, D12, D12, C2×C12, C22×S3, C5×S3, C30, C30, C2×D8, C40, C2×C20, C5×D4, C22×C10, D24, C2×C24, C2×D12, C60, S3×C10, C2×C30, C2×C40, C5×D8, D4×C10, C2×D24, C120, C5×D12, C5×D12, C2×C60, S3×C2×C10, C10×D8, C5×D24, C2×C120, C10×D12, C10×D24
Quotients: C1, C2, C22, C5, S3, D4, C23, C10, D6, D8, C2×D4, C2×C10, D12, C22×S3, C5×S3, C2×D8, C5×D4, C22×C10, D24, C2×D12, S3×C10, C5×D8, D4×C10, C2×D24, C5×D12, S3×C2×C10, C10×D8, C5×D24, C10×D12, C10×D24

Smallest permutation representation of C10×D24
On 240 points
Generators in S240
(1 67 27 166 107 193 130 80 181 231)(2 68 28 167 108 194 131 81 182 232)(3 69 29 168 109 195 132 82 183 233)(4 70 30 145 110 196 133 83 184 234)(5 71 31 146 111 197 134 84 185 235)(6 72 32 147 112 198 135 85 186 236)(7 49 33 148 113 199 136 86 187 237)(8 50 34 149 114 200 137 87 188 238)(9 51 35 150 115 201 138 88 189 239)(10 52 36 151 116 202 139 89 190 240)(11 53 37 152 117 203 140 90 191 217)(12 54 38 153 118 204 141 91 192 218)(13 55 39 154 119 205 142 92 169 219)(14 56 40 155 120 206 143 93 170 220)(15 57 41 156 97 207 144 94 171 221)(16 58 42 157 98 208 121 95 172 222)(17 59 43 158 99 209 122 96 173 223)(18 60 44 159 100 210 123 73 174 224)(19 61 45 160 101 211 124 74 175 225)(20 62 46 161 102 212 125 75 176 226)(21 63 47 162 103 213 126 76 177 227)(22 64 48 163 104 214 127 77 178 228)(23 65 25 164 105 215 128 78 179 229)(24 66 26 165 106 216 129 79 180 230)
(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)(217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 216)(2 215)(3 214)(4 213)(5 212)(6 211)(7 210)(8 209)(9 208)(10 207)(11 206)(12 205)(13 204)(14 203)(15 202)(16 201)(17 200)(18 199)(19 198)(20 197)(21 196)(22 195)(23 194)(24 193)(25 81)(26 80)(27 79)(28 78)(29 77)(30 76)(31 75)(32 74)(33 73)(34 96)(35 95)(36 94)(37 93)(38 92)(39 91)(40 90)(41 89)(42 88)(43 87)(44 86)(45 85)(46 84)(47 83)(48 82)(49 123)(50 122)(51 121)(52 144)(53 143)(54 142)(55 141)(56 140)(57 139)(58 138)(59 137)(60 136)(61 135)(62 134)(63 133)(64 132)(65 131)(66 130)(67 129)(68 128)(69 127)(70 126)(71 125)(72 124)(97 240)(98 239)(99 238)(100 237)(101 236)(102 235)(103 234)(104 233)(105 232)(106 231)(107 230)(108 229)(109 228)(110 227)(111 226)(112 225)(113 224)(114 223)(115 222)(116 221)(117 220)(118 219)(119 218)(120 217)(145 177)(146 176)(147 175)(148 174)(149 173)(150 172)(151 171)(152 170)(153 169)(154 192)(155 191)(156 190)(157 189)(158 188)(159 187)(160 186)(161 185)(162 184)(163 183)(164 182)(165 181)(166 180)(167 179)(168 178)

G:=sub<Sym(240)| (1,67,27,166,107,193,130,80,181,231)(2,68,28,167,108,194,131,81,182,232)(3,69,29,168,109,195,132,82,183,233)(4,70,30,145,110,196,133,83,184,234)(5,71,31,146,111,197,134,84,185,235)(6,72,32,147,112,198,135,85,186,236)(7,49,33,148,113,199,136,86,187,237)(8,50,34,149,114,200,137,87,188,238)(9,51,35,150,115,201,138,88,189,239)(10,52,36,151,116,202,139,89,190,240)(11,53,37,152,117,203,140,90,191,217)(12,54,38,153,118,204,141,91,192,218)(13,55,39,154,119,205,142,92,169,219)(14,56,40,155,120,206,143,93,170,220)(15,57,41,156,97,207,144,94,171,221)(16,58,42,157,98,208,121,95,172,222)(17,59,43,158,99,209,122,96,173,223)(18,60,44,159,100,210,123,73,174,224)(19,61,45,160,101,211,124,74,175,225)(20,62,46,161,102,212,125,75,176,226)(21,63,47,162,103,213,126,76,177,227)(22,64,48,163,104,214,127,77,178,228)(23,65,25,164,105,215,128,78,179,229)(24,66,26,165,106,216,129,79,180,230), (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)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,216)(2,215)(3,214)(4,213)(5,212)(6,211)(7,210)(8,209)(9,208)(10,207)(11,206)(12,205)(13,204)(14,203)(15,202)(16,201)(17,200)(18,199)(19,198)(20,197)(21,196)(22,195)(23,194)(24,193)(25,81)(26,80)(27,79)(28,78)(29,77)(30,76)(31,75)(32,74)(33,73)(34,96)(35,95)(36,94)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(43,87)(44,86)(45,85)(46,84)(47,83)(48,82)(49,123)(50,122)(51,121)(52,144)(53,143)(54,142)(55,141)(56,140)(57,139)(58,138)(59,137)(60,136)(61,135)(62,134)(63,133)(64,132)(65,131)(66,130)(67,129)(68,128)(69,127)(70,126)(71,125)(72,124)(97,240)(98,239)(99,238)(100,237)(101,236)(102,235)(103,234)(104,233)(105,232)(106,231)(107,230)(108,229)(109,228)(110,227)(111,226)(112,225)(113,224)(114,223)(115,222)(116,221)(117,220)(118,219)(119,218)(120,217)(145,177)(146,176)(147,175)(148,174)(149,173)(150,172)(151,171)(152,170)(153,169)(154,192)(155,191)(156,190)(157,189)(158,188)(159,187)(160,186)(161,185)(162,184)(163,183)(164,182)(165,181)(166,180)(167,179)(168,178)>;

G:=Group( (1,67,27,166,107,193,130,80,181,231)(2,68,28,167,108,194,131,81,182,232)(3,69,29,168,109,195,132,82,183,233)(4,70,30,145,110,196,133,83,184,234)(5,71,31,146,111,197,134,84,185,235)(6,72,32,147,112,198,135,85,186,236)(7,49,33,148,113,199,136,86,187,237)(8,50,34,149,114,200,137,87,188,238)(9,51,35,150,115,201,138,88,189,239)(10,52,36,151,116,202,139,89,190,240)(11,53,37,152,117,203,140,90,191,217)(12,54,38,153,118,204,141,91,192,218)(13,55,39,154,119,205,142,92,169,219)(14,56,40,155,120,206,143,93,170,220)(15,57,41,156,97,207,144,94,171,221)(16,58,42,157,98,208,121,95,172,222)(17,59,43,158,99,209,122,96,173,223)(18,60,44,159,100,210,123,73,174,224)(19,61,45,160,101,211,124,74,175,225)(20,62,46,161,102,212,125,75,176,226)(21,63,47,162,103,213,126,76,177,227)(22,64,48,163,104,214,127,77,178,228)(23,65,25,164,105,215,128,78,179,229)(24,66,26,165,106,216,129,79,180,230), (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)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,216)(2,215)(3,214)(4,213)(5,212)(6,211)(7,210)(8,209)(9,208)(10,207)(11,206)(12,205)(13,204)(14,203)(15,202)(16,201)(17,200)(18,199)(19,198)(20,197)(21,196)(22,195)(23,194)(24,193)(25,81)(26,80)(27,79)(28,78)(29,77)(30,76)(31,75)(32,74)(33,73)(34,96)(35,95)(36,94)(37,93)(38,92)(39,91)(40,90)(41,89)(42,88)(43,87)(44,86)(45,85)(46,84)(47,83)(48,82)(49,123)(50,122)(51,121)(52,144)(53,143)(54,142)(55,141)(56,140)(57,139)(58,138)(59,137)(60,136)(61,135)(62,134)(63,133)(64,132)(65,131)(66,130)(67,129)(68,128)(69,127)(70,126)(71,125)(72,124)(97,240)(98,239)(99,238)(100,237)(101,236)(102,235)(103,234)(104,233)(105,232)(106,231)(107,230)(108,229)(109,228)(110,227)(111,226)(112,225)(113,224)(114,223)(115,222)(116,221)(117,220)(118,219)(119,218)(120,217)(145,177)(146,176)(147,175)(148,174)(149,173)(150,172)(151,171)(152,170)(153,169)(154,192)(155,191)(156,190)(157,189)(158,188)(159,187)(160,186)(161,185)(162,184)(163,183)(164,182)(165,181)(166,180)(167,179)(168,178) );

G=PermutationGroup([[(1,67,27,166,107,193,130,80,181,231),(2,68,28,167,108,194,131,81,182,232),(3,69,29,168,109,195,132,82,183,233),(4,70,30,145,110,196,133,83,184,234),(5,71,31,146,111,197,134,84,185,235),(6,72,32,147,112,198,135,85,186,236),(7,49,33,148,113,199,136,86,187,237),(8,50,34,149,114,200,137,87,188,238),(9,51,35,150,115,201,138,88,189,239),(10,52,36,151,116,202,139,89,190,240),(11,53,37,152,117,203,140,90,191,217),(12,54,38,153,118,204,141,91,192,218),(13,55,39,154,119,205,142,92,169,219),(14,56,40,155,120,206,143,93,170,220),(15,57,41,156,97,207,144,94,171,221),(16,58,42,157,98,208,121,95,172,222),(17,59,43,158,99,209,122,96,173,223),(18,60,44,159,100,210,123,73,174,224),(19,61,45,160,101,211,124,74,175,225),(20,62,46,161,102,212,125,75,176,226),(21,63,47,162,103,213,126,76,177,227),(22,64,48,163,104,214,127,77,178,228),(23,65,25,164,105,215,128,78,179,229),(24,66,26,165,106,216,129,79,180,230)], [(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),(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,216),(2,215),(3,214),(4,213),(5,212),(6,211),(7,210),(8,209),(9,208),(10,207),(11,206),(12,205),(13,204),(14,203),(15,202),(16,201),(17,200),(18,199),(19,198),(20,197),(21,196),(22,195),(23,194),(24,193),(25,81),(26,80),(27,79),(28,78),(29,77),(30,76),(31,75),(32,74),(33,73),(34,96),(35,95),(36,94),(37,93),(38,92),(39,91),(40,90),(41,89),(42,88),(43,87),(44,86),(45,85),(46,84),(47,83),(48,82),(49,123),(50,122),(51,121),(52,144),(53,143),(54,142),(55,141),(56,140),(57,139),(58,138),(59,137),(60,136),(61,135),(62,134),(63,133),(64,132),(65,131),(66,130),(67,129),(68,128),(69,127),(70,126),(71,125),(72,124),(97,240),(98,239),(99,238),(100,237),(101,236),(102,235),(103,234),(104,233),(105,232),(106,231),(107,230),(108,229),(109,228),(110,227),(111,226),(112,225),(113,224),(114,223),(115,222),(116,221),(117,220),(118,219),(119,218),(120,217),(145,177),(146,176),(147,175),(148,174),(149,173),(150,172),(151,171),(152,170),(153,169),(154,192),(155,191),(156,190),(157,189),(158,188),(159,187),(160,186),(161,185),(162,184),(163,183),(164,182),(165,181),(166,180),(167,179),(168,178)]])

150 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 5A 5B 5C 5D 6A 6B 6C 8A 8B 8C 8D 10A ··· 10L 10M ··· 10AB 12A 12B 12C 12D 15A 15B 15C 15D 20A ··· 20H 24A ··· 24H 30A ··· 30L 40A ··· 40P 60A ··· 60P 120A ··· 120AF order 1 2 2 2 2 2 2 2 3 4 4 5 5 5 5 6 6 6 8 8 8 8 10 ··· 10 10 ··· 10 12 12 12 12 15 15 15 15 20 ··· 20 24 ··· 24 30 ··· 30 40 ··· 40 60 ··· 60 120 ··· 120 size 1 1 1 1 12 12 12 12 2 2 2 1 1 1 1 2 2 2 2 2 2 2 1 ··· 1 12 ··· 12 2 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

150 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 S3 D4 D4 D6 D6 D8 D12 D12 C5×S3 C5×D4 C5×D4 D24 S3×C10 S3×C10 C5×D8 C5×D12 C5×D12 C5×D24 kernel C10×D24 C5×D24 C2×C120 C10×D12 C2×D24 D24 C2×C24 C2×D12 C2×C40 C60 C2×C30 C40 C2×C20 C30 C20 C2×C10 C2×C8 C12 C2×C6 C10 C8 C2×C4 C6 C4 C22 C2 # reps 1 4 1 2 4 16 4 8 1 1 1 2 1 4 2 2 4 4 4 8 8 4 16 8 8 32

Matrix representation of C10×D24 in GL3(𝔽241) generated by

 240 0 0 0 150 0 0 0 150
,
 240 0 0 0 114 136 0 105 9
,
 240 0 0 0 127 9 0 136 114
G:=sub<GL(3,GF(241))| [240,0,0,0,150,0,0,0,150],[240,0,0,0,114,105,0,136,9],[240,0,0,0,127,136,0,9,114] >;

C10×D24 in GAP, Magma, Sage, TeX

C_{10}\times D_{24}
% in TeX

G:=Group("C10xD24");
// GroupNames label

G:=SmallGroup(480,782);
// by ID

G=gap.SmallGroup(480,782);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,926,646,4204,102,15686]);
// Polycyclic

G:=Group<a,b,c|a^10=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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