direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C2×C24, C120⋊44C22, C60.274C23, (C2×C40)⋊8C6, C40⋊10(C2×C6), C10⋊3(C2×C24), C30⋊11(C2×C8), C5⋊3(C22×C24), (C2×C120)⋊21C2, C15⋊13(C22×C8), C12.93(C4×D5), C4.23(D5×C12), C60.214(C2×C4), C20.48(C2×C12), (D5×C12).20C4, (C4×D5).10C12, D10.19(C2×C12), (C2×C12).449D10, C20.34(C22×C6), (C6×Dic5).32C4, C22.13(D5×C12), C10.25(C22×C12), (C2×C60).549C22, C30.183(C22×C4), (C2×Dic5).16C12, Dic5.22(C2×C12), (C22×D5).10C12, C12.241(C22×D5), (D5×C12).139C22, C4.35(D5×C2×C6), C2.2(D5×C2×C12), (C2×C4×D5).20C6, (D5×C2×C6).20C4, (C2×C5⋊2C8)⋊13C6, (C6×C5⋊2C8)⋊27C2, C6.108(C2×C4×D5), C5⋊2C8⋊13(C2×C6), (D5×C2×C12).41C2, (C2×C6).62(C4×D5), (C2×C4).98(C6×D5), (C4×D5).37(C2×C6), (C6×D5).68(C2×C4), (C2×C30).148(C2×C4), (C2×C10).33(C2×C12), (C2×C20).113(C2×C6), (C3×C5⋊2C8)⋊46C22, (C3×Dic5).77(C2×C4), SmallGroup(480,692)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C2×C24 |
Generators and relations for D5×C2×C24
G = < a,b,c,d | a2=b24=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 368 in 152 conjugacy classes, 98 normal (38 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C12, C12, C2×C6, C2×C6, C15, C2×C8, C2×C8, C22×C4, Dic5, C20, D10, C2×C10, C24, C24, C2×C12, C2×C12, C22×C6, C3×D5, C30, C30, C22×C8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C24, C2×C24, C22×C12, C3×Dic5, C60, C6×D5, C2×C30, C8×D5, C2×C5⋊2C8, C2×C40, C2×C4×D5, C22×C24, C3×C5⋊2C8, C120, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, D5×C2×C8, D5×C24, C6×C5⋊2C8, C2×C120, D5×C2×C12, D5×C2×C24
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, C23, D5, C12, C2×C6, C2×C8, C22×C4, D10, C24, C2×C12, C22×C6, C3×D5, C22×C8, C4×D5, C22×D5, C2×C24, C22×C12, C6×D5, C8×D5, C2×C4×D5, C22×C24, D5×C12, D5×C2×C6, D5×C2×C8, D5×C24, D5×C2×C12, D5×C2×C24
►On 240 points
Generators in S240
(1 165)(2 166)(3 167)(4 168)(5 145)(6 146)(7 147)(8 148)(9 149)(10 150)(11 151)(12 152)(13 153)(14 154)(15 155)(16 156)(17 157)(18 158)(19 159)(20 160)(21 161)(22 162)(23 163)(24 164)(25 95)(26 96)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 81)(36 82)(37 83)(38 84)(39 85)(40 86)(41 87)(42 88)(43 89)(44 90)(45 91)(46 92)(47 93)(48 94)(49 114)(50 115)(51 116)(52 117)(53 118)(54 119)(55 120)(56 97)(57 98)(58 99)(59 100)(60 101)(61 102)(62 103)(63 104)(64 105)(65 106)(66 107)(67 108)(68 109)(69 110)(70 111)(71 112)(72 113)(121 207)(122 208)(123 209)(124 210)(125 211)(126 212)(127 213)(128 214)(129 215)(130 216)(131 193)(132 194)(133 195)(134 196)(135 197)(136 198)(137 199)(138 200)(139 201)(140 202)(141 203)(142 204)(143 205)(144 206)(169 222)(170 223)(171 224)(172 225)(173 226)(174 227)(175 228)(176 229)(177 230)(178 231)(179 232)(180 233)(181 234)(182 235)(183 236)(184 237)(185 238)(186 239)(187 240)(188 217)(189 218)(190 219)(191 220)(192 221)
(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)
(1 171 97 90 130)(2 172 98 91 131)(3 173 99 92 132)(4 174 100 93 133)(5 175 101 94 134)(6 176 102 95 135)(7 177 103 96 136)(8 178 104 73 137)(9 179 105 74 138)(10 180 106 75 139)(11 181 107 76 140)(12 182 108 77 141)(13 183 109 78 142)(14 184 110 79 143)(15 185 111 80 144)(16 186 112 81 121)(17 187 113 82 122)(18 188 114 83 123)(19 189 115 84 124)(20 190 116 85 125)(21 191 117 86 126)(22 192 118 87 127)(23 169 119 88 128)(24 170 120 89 129)(25 197 146 229 61)(26 198 147 230 62)(27 199 148 231 63)(28 200 149 232 64)(29 201 150 233 65)(30 202 151 234 66)(31 203 152 235 67)(32 204 153 236 68)(33 205 154 237 69)(34 206 155 238 70)(35 207 156 239 71)(36 208 157 240 72)(37 209 158 217 49)(38 210 159 218 50)(39 211 160 219 51)(40 212 161 220 52)(41 213 162 221 53)(42 214 163 222 54)(43 215 164 223 55)(44 216 165 224 56)(45 193 166 225 57)(46 194 167 226 58)(47 195 168 227 59)(48 196 145 228 60)
(1 204)(2 205)(3 206)(4 207)(5 208)(6 209)(7 210)(8 211)(9 212)(10 213)(11 214)(12 215)(13 216)(14 193)(15 194)(16 195)(17 196)(18 197)(19 198)(20 199)(21 200)(22 201)(23 202)(24 203)(25 188)(26 189)(27 190)(28 191)(29 192)(30 169)(31 170)(32 171)(33 172)(34 173)(35 174)(36 175)(37 176)(38 177)(39 178)(40 179)(41 180)(42 181)(43 182)(44 183)(45 184)(46 185)(47 186)(48 187)(49 102)(50 103)(51 104)(52 105)(53 106)(54 107)(55 108)(56 109)(57 110)(58 111)(59 112)(60 113)(61 114)(62 115)(63 116)(64 117)(65 118)(66 119)(67 120)(68 97)(69 98)(70 99)(71 100)(72 101)(73 219)(74 220)(75 221)(76 222)(77 223)(78 224)(79 225)(80 226)(81 227)(82 228)(83 229)(84 230)(85 231)(86 232)(87 233)(88 234)(89 235)(90 236)(91 237)(92 238)(93 239)(94 240)(95 217)(96 218)(121 168)(122 145)(123 146)(124 147)(125 148)(126 149)(127 150)(128 151)(129 152)(130 153)(131 154)(132 155)(133 156)(134 157)(135 158)(136 159)(137 160)(138 161)(139 162)(140 163)(141 164)(142 165)(143 166)(144 167)
G:=sub<Sym(240)| (1,165)(2,166)(3,167)(4,168)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(21,161)(22,162)(23,163)(24,164)(25,95)(26,96)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,82)(37,83)(38,84)(39,85)(40,86)(41,87)(42,88)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(49,114)(50,115)(51,116)(52,117)(53,118)(54,119)(55,120)(56,97)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,113)(121,207)(122,208)(123,209)(124,210)(125,211)(126,212)(127,213)(128,214)(129,215)(130,216)(131,193)(132,194)(133,195)(134,196)(135,197)(136,198)(137,199)(138,200)(139,201)(140,202)(141,203)(142,204)(143,205)(144,206)(169,222)(170,223)(171,224)(172,225)(173,226)(174,227)(175,228)(176,229)(177,230)(178,231)(179,232)(180,233)(181,234)(182,235)(183,236)(184,237)(185,238)(186,239)(187,240)(188,217)(189,218)(190,219)(191,220)(192,221), (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), (1,171,97,90,130)(2,172,98,91,131)(3,173,99,92,132)(4,174,100,93,133)(5,175,101,94,134)(6,176,102,95,135)(7,177,103,96,136)(8,178,104,73,137)(9,179,105,74,138)(10,180,106,75,139)(11,181,107,76,140)(12,182,108,77,141)(13,183,109,78,142)(14,184,110,79,143)(15,185,111,80,144)(16,186,112,81,121)(17,187,113,82,122)(18,188,114,83,123)(19,189,115,84,124)(20,190,116,85,125)(21,191,117,86,126)(22,192,118,87,127)(23,169,119,88,128)(24,170,120,89,129)(25,197,146,229,61)(26,198,147,230,62)(27,199,148,231,63)(28,200,149,232,64)(29,201,150,233,65)(30,202,151,234,66)(31,203,152,235,67)(32,204,153,236,68)(33,205,154,237,69)(34,206,155,238,70)(35,207,156,239,71)(36,208,157,240,72)(37,209,158,217,49)(38,210,159,218,50)(39,211,160,219,51)(40,212,161,220,52)(41,213,162,221,53)(42,214,163,222,54)(43,215,164,223,55)(44,216,165,224,56)(45,193,166,225,57)(46,194,167,226,58)(47,195,168,227,59)(48,196,145,228,60), (1,204)(2,205)(3,206)(4,207)(5,208)(6,209)(7,210)(8,211)(9,212)(10,213)(11,214)(12,215)(13,216)(14,193)(15,194)(16,195)(17,196)(18,197)(19,198)(20,199)(21,200)(22,201)(23,202)(24,203)(25,188)(26,189)(27,190)(28,191)(29,192)(30,169)(31,170)(32,171)(33,172)(34,173)(35,174)(36,175)(37,176)(38,177)(39,178)(40,179)(41,180)(42,181)(43,182)(44,183)(45,184)(46,185)(47,186)(48,187)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,109)(57,110)(58,111)(59,112)(60,113)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,97)(69,98)(70,99)(71,100)(72,101)(73,219)(74,220)(75,221)(76,222)(77,223)(78,224)(79,225)(80,226)(81,227)(82,228)(83,229)(84,230)(85,231)(86,232)(87,233)(88,234)(89,235)(90,236)(91,237)(92,238)(93,239)(94,240)(95,217)(96,218)(121,168)(122,145)(123,146)(124,147)(125,148)(126,149)(127,150)(128,151)(129,152)(130,153)(131,154)(132,155)(133,156)(134,157)(135,158)(136,159)(137,160)(138,161)(139,162)(140,163)(141,164)(142,165)(143,166)(144,167)>;
G:=Group( (1,165)(2,166)(3,167)(4,168)(5,145)(6,146)(7,147)(8,148)(9,149)(10,150)(11,151)(12,152)(13,153)(14,154)(15,155)(16,156)(17,157)(18,158)(19,159)(20,160)(21,161)(22,162)(23,163)(24,164)(25,95)(26,96)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,79)(34,80)(35,81)(36,82)(37,83)(38,84)(39,85)(40,86)(41,87)(42,88)(43,89)(44,90)(45,91)(46,92)(47,93)(48,94)(49,114)(50,115)(51,116)(52,117)(53,118)(54,119)(55,120)(56,97)(57,98)(58,99)(59,100)(60,101)(61,102)(62,103)(63,104)(64,105)(65,106)(66,107)(67,108)(68,109)(69,110)(70,111)(71,112)(72,113)(121,207)(122,208)(123,209)(124,210)(125,211)(126,212)(127,213)(128,214)(129,215)(130,216)(131,193)(132,194)(133,195)(134,196)(135,197)(136,198)(137,199)(138,200)(139,201)(140,202)(141,203)(142,204)(143,205)(144,206)(169,222)(170,223)(171,224)(172,225)(173,226)(174,227)(175,228)(176,229)(177,230)(178,231)(179,232)(180,233)(181,234)(182,235)(183,236)(184,237)(185,238)(186,239)(187,240)(188,217)(189,218)(190,219)(191,220)(192,221), (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), (1,171,97,90,130)(2,172,98,91,131)(3,173,99,92,132)(4,174,100,93,133)(5,175,101,94,134)(6,176,102,95,135)(7,177,103,96,136)(8,178,104,73,137)(9,179,105,74,138)(10,180,106,75,139)(11,181,107,76,140)(12,182,108,77,141)(13,183,109,78,142)(14,184,110,79,143)(15,185,111,80,144)(16,186,112,81,121)(17,187,113,82,122)(18,188,114,83,123)(19,189,115,84,124)(20,190,116,85,125)(21,191,117,86,126)(22,192,118,87,127)(23,169,119,88,128)(24,170,120,89,129)(25,197,146,229,61)(26,198,147,230,62)(27,199,148,231,63)(28,200,149,232,64)(29,201,150,233,65)(30,202,151,234,66)(31,203,152,235,67)(32,204,153,236,68)(33,205,154,237,69)(34,206,155,238,70)(35,207,156,239,71)(36,208,157,240,72)(37,209,158,217,49)(38,210,159,218,50)(39,211,160,219,51)(40,212,161,220,52)(41,213,162,221,53)(42,214,163,222,54)(43,215,164,223,55)(44,216,165,224,56)(45,193,166,225,57)(46,194,167,226,58)(47,195,168,227,59)(48,196,145,228,60), (1,204)(2,205)(3,206)(4,207)(5,208)(6,209)(7,210)(8,211)(9,212)(10,213)(11,214)(12,215)(13,216)(14,193)(15,194)(16,195)(17,196)(18,197)(19,198)(20,199)(21,200)(22,201)(23,202)(24,203)(25,188)(26,189)(27,190)(28,191)(29,192)(30,169)(31,170)(32,171)(33,172)(34,173)(35,174)(36,175)(37,176)(38,177)(39,178)(40,179)(41,180)(42,181)(43,182)(44,183)(45,184)(46,185)(47,186)(48,187)(49,102)(50,103)(51,104)(52,105)(53,106)(54,107)(55,108)(56,109)(57,110)(58,111)(59,112)(60,113)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,97)(69,98)(70,99)(71,100)(72,101)(73,219)(74,220)(75,221)(76,222)(77,223)(78,224)(79,225)(80,226)(81,227)(82,228)(83,229)(84,230)(85,231)(86,232)(87,233)(88,234)(89,235)(90,236)(91,237)(92,238)(93,239)(94,240)(95,217)(96,218)(121,168)(122,145)(123,146)(124,147)(125,148)(126,149)(127,150)(128,151)(129,152)(130,153)(131,154)(132,155)(133,156)(134,157)(135,158)(136,159)(137,160)(138,161)(139,162)(140,163)(141,164)(142,165)(143,166)(144,167) );
G=PermutationGroup([[(1,165),(2,166),(3,167),(4,168),(5,145),(6,146),(7,147),(8,148),(9,149),(10,150),(11,151),(12,152),(13,153),(14,154),(15,155),(16,156),(17,157),(18,158),(19,159),(20,160),(21,161),(22,162),(23,163),(24,164),(25,95),(26,96),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,79),(34,80),(35,81),(36,82),(37,83),(38,84),(39,85),(40,86),(41,87),(42,88),(43,89),(44,90),(45,91),(46,92),(47,93),(48,94),(49,114),(50,115),(51,116),(52,117),(53,118),(54,119),(55,120),(56,97),(57,98),(58,99),(59,100),(60,101),(61,102),(62,103),(63,104),(64,105),(65,106),(66,107),(67,108),(68,109),(69,110),(70,111),(71,112),(72,113),(121,207),(122,208),(123,209),(124,210),(125,211),(126,212),(127,213),(128,214),(129,215),(130,216),(131,193),(132,194),(133,195),(134,196),(135,197),(136,198),(137,199),(138,200),(139,201),(140,202),(141,203),(142,204),(143,205),(144,206),(169,222),(170,223),(171,224),(172,225),(173,226),(174,227),(175,228),(176,229),(177,230),(178,231),(179,232),(180,233),(181,234),(182,235),(183,236),(184,237),(185,238),(186,239),(187,240),(188,217),(189,218),(190,219),(191,220),(192,221)], [(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)], [(1,171,97,90,130),(2,172,98,91,131),(3,173,99,92,132),(4,174,100,93,133),(5,175,101,94,134),(6,176,102,95,135),(7,177,103,96,136),(8,178,104,73,137),(9,179,105,74,138),(10,180,106,75,139),(11,181,107,76,140),(12,182,108,77,141),(13,183,109,78,142),(14,184,110,79,143),(15,185,111,80,144),(16,186,112,81,121),(17,187,113,82,122),(18,188,114,83,123),(19,189,115,84,124),(20,190,116,85,125),(21,191,117,86,126),(22,192,118,87,127),(23,169,119,88,128),(24,170,120,89,129),(25,197,146,229,61),(26,198,147,230,62),(27,199,148,231,63),(28,200,149,232,64),(29,201,150,233,65),(30,202,151,234,66),(31,203,152,235,67),(32,204,153,236,68),(33,205,154,237,69),(34,206,155,238,70),(35,207,156,239,71),(36,208,157,240,72),(37,209,158,217,49),(38,210,159,218,50),(39,211,160,219,51),(40,212,161,220,52),(41,213,162,221,53),(42,214,163,222,54),(43,215,164,223,55),(44,216,165,224,56),(45,193,166,225,57),(46,194,167,226,58),(47,195,168,227,59),(48,196,145,228,60)], [(1,204),(2,205),(3,206),(4,207),(5,208),(6,209),(7,210),(8,211),(9,212),(10,213),(11,214),(12,215),(13,216),(14,193),(15,194),(16,195),(17,196),(18,197),(19,198),(20,199),(21,200),(22,201),(23,202),(24,203),(25,188),(26,189),(27,190),(28,191),(29,192),(30,169),(31,170),(32,171),(33,172),(34,173),(35,174),(36,175),(37,176),(38,177),(39,178),(40,179),(41,180),(42,181),(43,182),(44,183),(45,184),(46,185),(47,186),(48,187),(49,102),(50,103),(51,104),(52,105),(53,106),(54,107),(55,108),(56,109),(57,110),(58,111),(59,112),(60,113),(61,114),(62,115),(63,116),(64,117),(65,118),(66,119),(67,120),(68,97),(69,98),(70,99),(71,100),(72,101),(73,219),(74,220),(75,221),(76,222),(77,223),(78,224),(79,225),(80,226),(81,227),(82,228),(83,229),(84,230),(85,231),(86,232),(87,233),(88,234),(89,235),(90,236),(91,237),(92,238),(93,239),(94,240),(95,217),(96,218),(121,168),(122,145),(123,146),(124,147),(125,148),(126,149),(127,150),(128,151),(129,152),(130,153),(131,154),(132,155),(133,156),(134,157),(135,158),(136,159),(137,160),(138,161),(139,162),(140,163),(141,164),(142,165),(143,166),(144,167)]])
192 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | ··· | 6F | 6G | ··· | 6N | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10F | 12A | ··· | 12H | 12I | ··· | 12P | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | ··· | 24P | 24Q | ··· | 24AF | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
192 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C6 | C8 | C12 | C12 | C12 | C24 | D5 | D10 | D10 | C3×D5 | C4×D5 | C4×D5 | C6×D5 | C6×D5 | C8×D5 | D5×C12 | D5×C12 | D5×C24 |
| kernel | D5×C2×C24 | D5×C24 | C6×C5⋊2C8 | C2×C120 | D5×C2×C12 | D5×C2×C8 | D5×C12 | C6×Dic5 | D5×C2×C6 | C8×D5 | C2×C5⋊2C8 | C2×C40 | C2×C4×D5 | C6×D5 | C4×D5 | C2×Dic5 | C22×D5 | D10 | C2×C24 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 8 | 2 | 2 | 2 | 16 | 8 | 4 | 4 | 32 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 4 | 16 | 8 | 8 | 32 |
Matrix representation of D5×C2×C24 ►in GL4(𝔽241) generated by
| 240 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 |
| 0 | 0 | 225 | 0 |
| 0 | 0 | 0 | 225 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 1 |
| 0 | 0 | 50 | 190 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 50 | 1 |
G:=sub<GL(4,GF(241))| [240,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,30,0,0,0,0,225,0,0,0,0,225],[1,0,0,0,0,1,0,0,0,0,240,50,0,0,1,190],[1,0,0,0,0,1,0,0,0,0,240,50,0,0,0,1] >;
D5×C2×C24 in GAP, Magma, Sage, TeX
D_5\times C_2\times C_{24} % in TeX
G:=Group("D5xC2xC24"); // GroupNames label
G:=SmallGroup(480,692);
// by ID
G=gap.SmallGroup(480,692);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,142,102,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^24=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations