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G = C10×C24⋊C2order 480 = 25·3·5

Direct product of C10 and C24⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C10×C24⋊C2
 Chief series C1 — C3 — C6 — C12 — C60 — C5×D12 — C10×D12 — C10×C24⋊C2
 Lower central C3 — C6 — C12 — C10×C24⋊C2
 Upper central C1 — C2×C10 — C2×C20 — C2×C40

Generators and relations for C10×C24⋊C2
G = < a,b,c | a10=b24=c2=1, ab=ba, ac=ca, cbc=b11 >

Subgroups: 388 in 136 conjugacy classes, 66 normal (34 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×4], C5, S3 [×2], C6, C6 [×2], C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, C10, C10 [×2], C10 [×2], Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C15, C2×C8, SD16 [×4], C2×D4, C2×Q8, C20 [×2], C20 [×2], C2×C10, C2×C10 [×4], C24 [×2], Dic6 [×2], Dic6, D12 [×2], D12, C2×Dic3, C2×C12, C22×S3, C5×S3 [×2], C30, C30 [×2], C2×SD16, C40 [×2], C2×C20, C2×C20, C5×D4 [×3], C5×Q8 [×3], C22×C10, C24⋊C2 [×4], C2×C24, C2×Dic6, C2×D12, C5×Dic3 [×2], C60 [×2], S3×C10 [×4], C2×C30, C2×C40, C5×SD16 [×4], D4×C10, Q8×C10, C2×C24⋊C2, C120 [×2], C5×Dic6 [×2], C5×Dic6, C5×D12 [×2], C5×D12, C10×Dic3, C2×C60, S3×C2×C10, C10×SD16, C5×C24⋊C2 [×4], C2×C120, C10×Dic6, C10×D12, C10×C24⋊C2
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], SD16 [×2], C2×D4, C2×C10 [×7], D12 [×2], C22×S3, C5×S3, C2×SD16, C5×D4 [×2], C22×C10, C24⋊C2 [×2], C2×D12, S3×C10 [×3], C5×SD16 [×2], D4×C10, C2×C24⋊C2, C5×D12 [×2], S3×C2×C10, C10×SD16, C5×C24⋊C2 [×2], C10×D12, C10×C24⋊C2

Smallest permutation representation of C10×C24⋊C2
On 240 points
Generators in S240
(1 122 210 88 173 59 105 41 158 237)(2 123 211 89 174 60 106 42 159 238)(3 124 212 90 175 61 107 43 160 239)(4 125 213 91 176 62 108 44 161 240)(5 126 214 92 177 63 109 45 162 217)(6 127 215 93 178 64 110 46 163 218)(7 128 216 94 179 65 111 47 164 219)(8 129 193 95 180 66 112 48 165 220)(9 130 194 96 181 67 113 25 166 221)(10 131 195 73 182 68 114 26 167 222)(11 132 196 74 183 69 115 27 168 223)(12 133 197 75 184 70 116 28 145 224)(13 134 198 76 185 71 117 29 146 225)(14 135 199 77 186 72 118 30 147 226)(15 136 200 78 187 49 119 31 148 227)(16 137 201 79 188 50 120 32 149 228)(17 138 202 80 189 51 97 33 150 229)(18 139 203 81 190 52 98 34 151 230)(19 140 204 82 191 53 99 35 152 231)(20 141 205 83 192 54 100 36 153 232)(21 142 206 84 169 55 101 37 154 233)(22 143 207 85 170 56 102 38 155 234)(23 144 208 86 171 57 103 39 156 235)(24 121 209 87 172 58 104 40 157 236)
(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 59)(2 70)(3 57)(4 68)(5 55)(6 66)(7 53)(8 64)(9 51)(10 62)(11 49)(12 60)(13 71)(14 58)(15 69)(16 56)(17 67)(18 54)(19 65)(20 52)(21 63)(22 50)(23 61)(24 72)(25 202)(26 213)(27 200)(28 211)(29 198)(30 209)(31 196)(32 207)(33 194)(34 205)(35 216)(36 203)(37 214)(38 201)(39 212)(40 199)(41 210)(42 197)(43 208)(44 195)(45 206)(46 193)(47 204)(48 215)(73 161)(74 148)(75 159)(76 146)(77 157)(78 168)(79 155)(80 166)(81 153)(82 164)(83 151)(84 162)(85 149)(86 160)(87 147)(88 158)(89 145)(90 156)(91 167)(92 154)(93 165)(94 152)(95 163)(96 150)(97 130)(98 141)(99 128)(100 139)(101 126)(102 137)(103 124)(104 135)(105 122)(106 133)(107 144)(108 131)(109 142)(110 129)(111 140)(112 127)(113 138)(114 125)(115 136)(116 123)(117 134)(118 121)(119 132)(120 143)(169 217)(170 228)(171 239)(172 226)(173 237)(174 224)(175 235)(176 222)(177 233)(178 220)(179 231)(180 218)(181 229)(182 240)(183 227)(184 238)(185 225)(186 236)(187 223)(188 234)(189 221)(190 232)(191 219)(192 230)

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

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

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

150 conjugacy classes

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

150 irreducible representations

 dim 1 1 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 C2 C5 C10 C10 C10 C10 S3 D4 D4 D6 D6 SD16 D12 D12 C5×S3 C5×D4 C5×D4 C24⋊C2 S3×C10 S3×C10 C5×SD16 C5×D12 C5×D12 C5×C24⋊C2 kernel C10×C24⋊C2 C5×C24⋊C2 C2×C120 C10×Dic6 C10×D12 C2×C24⋊C2 C24⋊C2 C2×C24 C2×Dic6 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 1 1 4 16 4 4 4 1 1 1 2 1 4 2 2 4 4 4 8 8 4 16 8 8 32

Matrix representation of C10×C24⋊C2 in GL3(𝔽241) generated by

 240 0 0 0 36 0 0 0 36
,
 240 0 0 0 213 175 0 66 147
,
 1 0 0 0 240 1 0 0 1
G:=sub<GL(3,GF(241))| [240,0,0,0,36,0,0,0,36],[240,0,0,0,213,66,0,175,147],[1,0,0,0,240,0,0,1,1] >;

C10×C24⋊C2 in GAP, Magma, Sage, TeX

C_{10}\times C_{24}\rtimes C_2
% in TeX

G:=Group("C10xC24:C2");
// GroupNames label

G:=SmallGroup(480,781);
// by ID

G=gap.SmallGroup(480,781);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,926,226,4204,102,15686]);
// 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^11>;
// generators/relations

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