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G = C5×C4○D24order 480 = 25·3·5

Direct product of C5 and C4○D24

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C5×C4○D24, D247C10, C40.78D6, C20.72D12, C60.192D4, Dic127C10, C60.269C23, C120.96C22, (C2×C24)⋊6C10, (C2×C40)⋊12S3, C24⋊C27C10, (C2×C120)⋊18C2, C4○D121C10, (C5×D24)⋊15C2, C1527(C4○D8), C8.17(S3×C10), C6.11(D4×C10), C12.35(C5×D4), C4.20(C5×D12), C24.17(C2×C10), D12.7(C2×C10), C30.298(C2×D4), (C2×C20).434D6, (C2×C10).12D12, (C2×C30).125D4, C2.13(C10×D12), C10.82(C2×D12), C22.1(C5×D12), (C5×Dic12)⋊15C2, Dic6.6(C2×C10), C12.30(C22×C10), (C2×C60).537C22, C20.233(C22×S3), (C5×D12).46C22, (C5×Dic6).48C22, C31(C5×C4○D8), (C2×C8)⋊4(C5×S3), C4.30(S3×C2×C10), (C2×C6).18(C5×D4), (C5×C24⋊C2)⋊15C2, (C5×C4○D12)⋊11C2, (C2×C4).82(S3×C10), (C2×C12).104(C2×C10), SmallGroup(480,783)

Series: Derived Chief Lower central Upper central

C1C12 — C5×C4○D24
C1C3C6C12C60C5×D12C5×C4○D12 — C5×C4○D24
C3C6C12 — C5×C4○D24
C1C20C2×C20C2×C40

Generators and relations for C5×C4○D24
 G = < a,b,c,d | a5=b4=d2=1, c12=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c11 >

Subgroups: 324 in 124 conjugacy classes, 58 normal (42 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], C5, S3 [×2], C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], C10, C10 [×3], Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C15, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C24 [×2], Dic6 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C2×C12, C5×S3 [×2], C30, C30, C4○D8, C40 [×2], C2×C20, C2×C20 [×2], C5×D4 [×4], C5×Q8 [×2], C24⋊C2 [×2], D24, Dic12, C2×C24, C4○D12 [×2], C5×Dic3 [×2], C60 [×2], S3×C10 [×2], C2×C30, C2×C40, C5×D8, C5×SD16 [×2], C5×Q16, C5×C4○D4 [×2], C4○D24, C120 [×2], C5×Dic6 [×2], S3×C20 [×2], C5×D12 [×2], C5×C3⋊D4 [×2], C2×C60, C5×C4○D8, C5×C24⋊C2 [×2], C5×D24, C5×Dic12, C2×C120, C5×C4○D12 [×2], C5×C4○D24
Quotients: C1, C2 [×7], C22 [×7], C5, S3, D4 [×2], C23, C10 [×7], D6 [×3], C2×D4, C2×C10 [×7], D12 [×2], C22×S3, C5×S3, C4○D8, C5×D4 [×2], C22×C10, C2×D12, S3×C10 [×3], D4×C10, C4○D24, C5×D12 [×2], S3×C2×C10, C5×C4○D8, C10×D12, C5×C4○D24

Smallest permutation representation of C5×C4○D24
On 240 points
Generators in S240
(1 151 60 173 80)(2 152 61 174 81)(3 153 62 175 82)(4 154 63 176 83)(5 155 64 177 84)(6 156 65 178 85)(7 157 66 179 86)(8 158 67 180 87)(9 159 68 181 88)(10 160 69 182 89)(11 161 70 183 90)(12 162 71 184 91)(13 163 72 185 92)(14 164 49 186 93)(15 165 50 187 94)(16 166 51 188 95)(17 167 52 189 96)(18 168 53 190 73)(19 145 54 191 74)(20 146 55 192 75)(21 147 56 169 76)(22 148 57 170 77)(23 149 58 171 78)(24 150 59 172 79)(25 115 121 230 211)(26 116 122 231 212)(27 117 123 232 213)(28 118 124 233 214)(29 119 125 234 215)(30 120 126 235 216)(31 97 127 236 193)(32 98 128 237 194)(33 99 129 238 195)(34 100 130 239 196)(35 101 131 240 197)(36 102 132 217 198)(37 103 133 218 199)(38 104 134 219 200)(39 105 135 220 201)(40 106 136 221 202)(41 107 137 222 203)(42 108 138 223 204)(43 109 139 224 205)(44 110 140 225 206)(45 111 141 226 207)(46 112 142 227 208)(47 113 143 228 209)(48 114 144 229 210)
(1 19 13 7)(2 20 14 8)(3 21 15 9)(4 22 16 10)(5 23 17 11)(6 24 18 12)(25 31 37 43)(26 32 38 44)(27 33 39 45)(28 34 40 46)(29 35 41 47)(30 36 42 48)(49 67 61 55)(50 68 62 56)(51 69 63 57)(52 70 64 58)(53 71 65 59)(54 72 66 60)(73 91 85 79)(74 92 86 80)(75 93 87 81)(76 94 88 82)(77 95 89 83)(78 96 90 84)(97 103 109 115)(98 104 110 116)(99 105 111 117)(100 106 112 118)(101 107 113 119)(102 108 114 120)(121 127 133 139)(122 128 134 140)(123 129 135 141)(124 130 136 142)(125 131 137 143)(126 132 138 144)(145 163 157 151)(146 164 158 152)(147 165 159 153)(148 166 160 154)(149 167 161 155)(150 168 162 156)(169 187 181 175)(170 188 182 176)(171 189 183 177)(172 190 184 178)(173 191 185 179)(174 192 186 180)(193 199 205 211)(194 200 206 212)(195 201 207 213)(196 202 208 214)(197 203 209 215)(198 204 210 216)(217 223 229 235)(218 224 230 236)(219 225 231 237)(220 226 232 238)(221 227 233 239)(222 228 234 240)
(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)
(1 133)(2 132)(3 131)(4 130)(5 129)(6 128)(7 127)(8 126)(9 125)(10 124)(11 123)(12 122)(13 121)(14 144)(15 143)(16 142)(17 141)(18 140)(19 139)(20 138)(21 137)(22 136)(23 135)(24 134)(25 185)(26 184)(27 183)(28 182)(29 181)(30 180)(31 179)(32 178)(33 177)(34 176)(35 175)(36 174)(37 173)(38 172)(39 171)(40 170)(41 169)(42 192)(43 191)(44 190)(45 189)(46 188)(47 187)(48 186)(49 210)(50 209)(51 208)(52 207)(53 206)(54 205)(55 204)(56 203)(57 202)(58 201)(59 200)(60 199)(61 198)(62 197)(63 196)(64 195)(65 194)(66 193)(67 216)(68 215)(69 214)(70 213)(71 212)(72 211)(73 110)(74 109)(75 108)(76 107)(77 106)(78 105)(79 104)(80 103)(81 102)(82 101)(83 100)(84 99)(85 98)(86 97)(87 120)(88 119)(89 118)(90 117)(91 116)(92 115)(93 114)(94 113)(95 112)(96 111)(145 224)(146 223)(147 222)(148 221)(149 220)(150 219)(151 218)(152 217)(153 240)(154 239)(155 238)(156 237)(157 236)(158 235)(159 234)(160 233)(161 232)(162 231)(163 230)(164 229)(165 228)(166 227)(167 226)(168 225)

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

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

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

150 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E5A5B5C5D6A6B6C8A8B8C8D10A10B10C10D10E10F10G10H10I···10P12A12B12C12D15A15B15C15D20A···20H20I20J20K20L20M···20T24A···24H30A···30L40A···40P60A···60P120A···120AF
order1222234444455556668888101010101010101010···10121212121515151520···202020202020···2024···2430···3040···4060···60120···120
size112121221121212111122222221111222212···12222222221···1222212···122···22···22···22···22···2

150 irreducible representations

dim111111111111222222222222222222
type+++++++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10S3D4D4D6D6D12D12C5×S3C4○D8C5×D4C5×D4S3×C10S3×C10C4○D24C5×D12C5×D12C5×C4○D8C5×C4○D24
kernelC5×C4○D24C5×C24⋊C2C5×D24C5×Dic12C2×C120C5×C4○D12C4○D24C24⋊C2D24Dic12C2×C24C4○D12C2×C40C60C2×C30C40C2×C20C20C2×C10C2×C8C15C12C2×C6C8C2×C4C5C4C22C3C1
# reps12111248444811121224444848881632

Matrix representation of C5×C4○D24 in GL2(𝔽241) generated by

910
091
,
1770
0177
,
2090
23128
,
113105
23128
G:=sub<GL(2,GF(241))| [91,0,0,91],[177,0,0,177],[209,23,0,128],[113,23,105,128] >;

C5×C4○D24 in GAP, Magma, Sage, TeX

C_5\times C_4\circ D_{24}
% in TeX

G:=Group("C5xC4oD24");
// GroupNames label

G:=SmallGroup(480,783);
// by ID

G=gap.SmallGroup(480,783);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,926,226,4204,102,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=d^2=1,c^12=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^11>;
// generators/relations

׿
×
𝔽