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## G = C5×D24⋊C2order 480 = 25·3·5

### Direct product of C5 and D24⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×D24⋊C2
 Chief series C1 — C3 — C6 — C12 — C60 — S3×C20 — C5×Q8⋊3S3 — C5×D24⋊C2
 Lower central C3 — C6 — C12 — C5×D24⋊C2
 Upper central C1 — C10 — C20 — C5×Q16

Generators and relations for C5×D24⋊C2
G = < a,b,c,d | a5=b24=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b17, dcd=b4c >

Subgroups: 324 in 124 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C5, S3 [×3], C6, C8, C8, C2×C4 [×3], D4 [×4], Q8 [×2], C10, C10 [×3], Dic3, C12, C12 [×2], D6, D6 [×2], C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C20, C20 [×3], C2×C10 [×3], C3⋊C8, C24, C4×S3, C4×S3 [×2], D12 [×2], D12 [×2], C3×Q8 [×2], C5×S3 [×3], C30, C4○D8, C40, C40, C2×C20 [×3], C5×D4 [×4], C5×Q8 [×2], S3×C8, D24, Q82S3 [×2], C3×Q16, Q83S3 [×2], C5×Dic3, C60, C60 [×2], S3×C10, S3×C10 [×2], C2×C40, C5×D8, C5×SD16 [×2], C5×Q16, C5×C4○D4 [×2], D24⋊C2, C5×C3⋊C8, C120, S3×C20, S3×C20 [×2], C5×D12 [×2], C5×D12 [×2], Q8×C15 [×2], C5×C4○D8, S3×C40, C5×D24, C5×Q82S3 [×2], C15×Q16, C5×Q83S3 [×2], C5×D24⋊C2
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×C10 [×7], C22×S3, C5×S3, C4○D8, C5×D4 [×2], C22×C10, S3×D4, S3×C10 [×3], D4×C10, D24⋊C2, S3×C2×C10, C5×C4○D8, C5×S3×D4, C5×D24⋊C2

Smallest permutation representation of C5×D24⋊C2
On 240 points
Generators in S240
(1 65 211 143 80)(2 66 212 144 81)(3 67 213 121 82)(4 68 214 122 83)(5 69 215 123 84)(6 70 216 124 85)(7 71 193 125 86)(8 72 194 126 87)(9 49 195 127 88)(10 50 196 128 89)(11 51 197 129 90)(12 52 198 130 91)(13 53 199 131 92)(14 54 200 132 93)(15 55 201 133 94)(16 56 202 134 95)(17 57 203 135 96)(18 58 204 136 73)(19 59 205 137 74)(20 60 206 138 75)(21 61 207 139 76)(22 62 208 140 77)(23 63 209 141 78)(24 64 210 142 79)(25 189 222 148 111)(26 190 223 149 112)(27 191 224 150 113)(28 192 225 151 114)(29 169 226 152 115)(30 170 227 153 116)(31 171 228 154 117)(32 172 229 155 118)(33 173 230 156 119)(34 174 231 157 120)(35 175 232 158 97)(36 176 233 159 98)(37 177 234 160 99)(38 178 235 161 100)(39 179 236 162 101)(40 180 237 163 102)(41 181 238 164 103)(42 182 239 165 104)(43 183 240 166 105)(44 184 217 167 106)(45 185 218 168 107)(46 186 219 145 108)(47 187 220 146 109)(48 188 221 147 110)
(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)
(1 148)(2 147)(3 146)(4 145)(5 168)(6 167)(7 166)(8 165)(9 164)(10 163)(11 162)(12 161)(13 160)(14 159)(15 158)(16 157)(17 156)(18 155)(19 154)(20 153)(21 152)(22 151)(23 150)(24 149)(25 211)(26 210)(27 209)(28 208)(29 207)(30 206)(31 205)(32 204)(33 203)(34 202)(35 201)(36 200)(37 199)(38 198)(39 197)(40 196)(41 195)(42 194)(43 193)(44 216)(45 215)(46 214)(47 213)(48 212)(49 103)(50 102)(51 101)(52 100)(53 99)(54 98)(55 97)(56 120)(57 119)(58 118)(59 117)(60 116)(61 115)(62 114)(63 113)(64 112)(65 111)(66 110)(67 109)(68 108)(69 107)(70 106)(71 105)(72 104)(73 229)(74 228)(75 227)(76 226)(77 225)(78 224)(79 223)(80 222)(81 221)(82 220)(83 219)(84 218)(85 217)(86 240)(87 239)(88 238)(89 237)(90 236)(91 235)(92 234)(93 233)(94 232)(95 231)(96 230)(121 187)(122 186)(123 185)(124 184)(125 183)(126 182)(127 181)(128 180)(129 179)(130 178)(131 177)(132 176)(133 175)(134 174)(135 173)(136 172)(137 171)(138 170)(139 169)(140 192)(141 191)(142 190)(143 189)(144 188)
(2 18)(3 11)(5 21)(6 14)(8 24)(9 17)(12 20)(15 23)(25 45)(26 38)(27 31)(28 48)(29 41)(30 34)(32 44)(33 37)(35 47)(36 40)(39 43)(42 46)(49 57)(51 67)(52 60)(54 70)(55 63)(58 66)(61 69)(64 72)(73 81)(75 91)(76 84)(78 94)(79 87)(82 90)(85 93)(88 96)(97 109)(98 102)(99 119)(100 112)(101 105)(103 115)(104 108)(106 118)(107 111)(110 114)(113 117)(116 120)(121 129)(123 139)(124 132)(126 142)(127 135)(130 138)(133 141)(136 144)(145 165)(146 158)(147 151)(148 168)(149 161)(150 154)(152 164)(153 157)(155 167)(156 160)(159 163)(162 166)(169 181)(170 174)(171 191)(172 184)(173 177)(175 187)(176 180)(178 190)(179 183)(182 186)(185 189)(188 192)(194 210)(195 203)(197 213)(198 206)(200 216)(201 209)(204 212)(207 215)(217 229)(218 222)(219 239)(220 232)(221 225)(223 235)(224 228)(226 238)(227 231)(230 234)(233 237)(236 240)

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

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

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

105 conjugacy classes

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

105 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 S3 D4 D4 D6 D6 C5×S3 C4○D8 C5×D4 C5×D4 S3×C10 S3×C10 C5×C4○D8 S3×D4 D24⋊C2 C5×S3×D4 C5×D24⋊C2 kernel C5×D24⋊C2 S3×C40 C5×D24 C5×Q8⋊2S3 C15×Q16 C5×Q8⋊3S3 D24⋊C2 S3×C8 D24 Q8⋊2S3 C3×Q16 Q8⋊3S3 C5×Q16 C5×Dic3 S3×C10 C40 C5×Q8 Q16 C15 Dic3 D6 C8 Q8 C3 C10 C5 C2 C1 # reps 1 1 1 2 1 2 4 4 4 8 4 8 1 1 1 1 2 4 4 4 4 4 8 16 1 2 4 8

Matrix representation of C5×D24⋊C2 in GL4(𝔽241) generated by

 87 0 0 0 0 87 0 0 0 0 1 0 0 0 0 1
,
 1 1 0 0 240 0 0 0 0 0 8 19 0 0 0 211
,
 1 1 0 0 0 240 0 0 0 0 233 222 0 0 16 8
,
 1 0 0 0 240 240 0 0 0 0 1 1 0 0 0 240
G:=sub<GL(4,GF(241))| [87,0,0,0,0,87,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,1,0,0,0,0,0,8,0,0,0,19,211],[1,0,0,0,1,240,0,0,0,0,233,16,0,0,222,8],[1,240,0,0,0,240,0,0,0,0,1,0,0,0,1,240] >;

C5×D24⋊C2 in GAP, Magma, Sage, TeX

C_5\times D_{24}\rtimes C_2
% in TeX

G:=Group("C5xD24:C2");
// GroupNames label

G:=SmallGroup(480,798);
// by ID

G=gap.SmallGroup(480,798);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,568,1766,471,436,2111,1068,102,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^24=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d=b^17,d*c*d=b^4*c>;
// generators/relations

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