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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 [×2], C2 [×4], C3, C4 [×2], C22, C22 [×8], C5, S3 [×4], C6, C6 [×2], C8 [×2], C2×C4, D4 [×6], C23 [×2], C10, C10 [×2], C10 [×4], C12 [×2], D6 [×8], C2×C6, C15, C2×C8, D8 [×4], C2×D4 [×2], C20 [×2], C2×C10, C2×C10 [×8], C24 [×2], D12 [×4], D12 [×2], C2×C12, C22×S3 [×2], C5×S3 [×4], C30, C30 [×2], C2×D8, C40 [×2], C2×C20, C5×D4 [×6], C22×C10 [×2], D24 [×4], C2×C24, C2×D12 [×2], C60 [×2], S3×C10 [×8], C2×C30, C2×C40, C5×D8 [×4], D4×C10 [×2], C2×D24, C120 [×2], C5×D12 [×4], C5×D12 [×2], C2×C60, S3×C2×C10 [×2], C10×D8, C5×D24 [×4], C2×C120, C10×D12 [×2], C10×D24
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], D8 [×2], C2×D4, C2×C10 [×7], D12 [×2], C22×S3, C5×S3, C2×D8, C5×D4 [×2], C22×C10, D24 [×2], C2×D12, S3×C10 [×3], C5×D8 [×2], D4×C10, C2×D24, C5×D12 [×2], S3×C2×C10, C10×D8, C5×D24 [×2], C10×D12, C10×D24

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

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

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

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

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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