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## G = C5×C8.D6order 480 = 25·3·5

### Direct product of C5 and C8.D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C5×C8.D6
 Chief series C1 — C3 — C6 — C12 — C60 — C5×D12 — C5×C4○D12 — C5×C8.D6
 Lower central C3 — C6 — C12 — C5×C8.D6
 Upper central C1 — C10 — C2×C20 — C5×M4(2)

Generators and relations for C5×C8.D6
G = < a,b,c,d | a5=b8=1, c6=d2=b4, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b-1, dcd-1=c5 >

Subgroups: 292 in 120 conjugacy classes, 58 normal (38 characteristic)
C1, C2, C2 [×2], C3, C4 [×2], C4 [×3], C22, C22, C5, S3, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], C10, C10 [×2], Dic3 [×3], C12 [×2], D6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, C20 [×2], C20 [×3], C2×C10, C2×C10, C24 [×2], Dic6, Dic6 [×2], Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C5×S3, C30, C30, C8.C22, C40 [×2], C2×C20, C2×C20 [×2], C5×D4 [×2], C5×Q8 [×4], C24⋊C2 [×2], Dic12 [×2], C3×M4(2), C2×Dic6, C4○D12, C5×Dic3 [×3], C60 [×2], S3×C10, C2×C30, C5×M4(2), C5×SD16 [×2], C5×Q16 [×2], Q8×C10, C5×C4○D4, C8.D6, C120 [×2], C5×Dic6, C5×Dic6 [×2], C5×Dic6, S3×C20, C5×D12, C10×Dic3, C5×C3⋊D4, C2×C60, C5×C8.C22, C5×C24⋊C2 [×2], C5×Dic12 [×2], C15×M4(2), C10×Dic6, C5×C4○D12, C5×C8.D6
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, C8.C22, C5×D4 [×2], C22×C10, C2×D12, S3×C10 [×3], D4×C10, C8.D6, C5×D12 [×2], S3×C2×C10, C5×C8.C22, C10×D12, C5×C8.D6

Smallest permutation representation of C5×C8.D6
On 240 points
Generators in S240
(1 54 44 32 22)(2 55 45 33 23)(3 56 46 34 24)(4 57 47 35 13)(5 58 48 36 14)(6 59 37 25 15)(7 60 38 26 16)(8 49 39 27 17)(9 50 40 28 18)(10 51 41 29 19)(11 52 42 30 20)(12 53 43 31 21)(61 113 102 85 82)(62 114 103 86 83)(63 115 104 87 84)(64 116 105 88 73)(65 117 106 89 74)(66 118 107 90 75)(67 119 108 91 76)(68 120 97 92 77)(69 109 98 93 78)(70 110 99 94 79)(71 111 100 95 80)(72 112 101 96 81)(121 175 157 154 133)(122 176 158 155 134)(123 177 159 156 135)(124 178 160 145 136)(125 179 161 146 137)(126 180 162 147 138)(127 169 163 148 139)(128 170 164 149 140)(129 171 165 150 141)(130 172 166 151 142)(131 173 167 152 143)(132 174 168 153 144)(181 229 220 205 196)(182 230 221 206 197)(183 231 222 207 198)(184 232 223 208 199)(185 233 224 209 200)(186 234 225 210 201)(187 235 226 211 202)(188 236 227 212 203)(189 237 228 213 204)(190 238 217 214 193)(191 239 218 215 194)(192 240 219 216 195)
(1 66 4 63 7 72 10 69)(2 61 5 70 8 67 11 64)(3 68 6 65 9 62 12 71)(13 84 16 81 19 78 22 75)(14 79 17 76 20 73 23 82)(15 74 18 83 21 80 24 77)(25 89 28 86 31 95 34 92)(26 96 29 93 32 90 35 87)(27 91 30 88 33 85 36 94)(37 106 40 103 43 100 46 97)(38 101 41 98 44 107 47 104)(39 108 42 105 45 102 48 99)(49 119 52 116 55 113 58 110)(50 114 53 111 56 120 59 117)(51 109 54 118 57 115 60 112)(121 181 130 184 127 187 124 190)(122 188 131 191 128 182 125 185)(123 183 132 186 129 189 126 192)(133 196 142 199 139 202 136 193)(134 203 143 194 140 197 137 200)(135 198 144 201 141 204 138 195)(145 214 154 205 151 208 148 211)(146 209 155 212 152 215 149 206)(147 216 156 207 153 210 150 213)(157 220 166 223 163 226 160 217)(158 227 167 218 164 221 161 224)(159 222 168 225 165 228 162 219)(169 235 178 238 175 229 172 232)(170 230 179 233 176 236 173 239)(171 237 180 240 177 231 174 234)
(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)
(1 123 7 129)(2 128 8 122)(3 121 9 127)(4 126 10 132)(5 131 11 125)(6 124 12 130)(13 138 19 144)(14 143 20 137)(15 136 21 142)(16 141 22 135)(17 134 23 140)(18 139 24 133)(25 145 31 151)(26 150 32 156)(27 155 33 149)(28 148 34 154)(29 153 35 147)(30 146 36 152)(37 160 43 166)(38 165 44 159)(39 158 45 164)(40 163 46 157)(41 168 47 162)(42 161 48 167)(49 176 55 170)(50 169 56 175)(51 174 57 180)(52 179 58 173)(53 172 59 178)(54 177 60 171)(61 191 67 185)(62 184 68 190)(63 189 69 183)(64 182 70 188)(65 187 71 181)(66 192 72 186)(73 197 79 203)(74 202 80 196)(75 195 81 201)(76 200 82 194)(77 193 83 199)(78 198 84 204)(85 215 91 209)(86 208 92 214)(87 213 93 207)(88 206 94 212)(89 211 95 205)(90 216 96 210)(97 217 103 223)(98 222 104 228)(99 227 105 221)(100 220 106 226)(101 225 107 219)(102 218 108 224)(109 231 115 237)(110 236 116 230)(111 229 117 235)(112 234 118 240)(113 239 119 233)(114 232 120 238)

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

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

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

105 conjugacy classes

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 5A 5B 5C 5D 6A 6B 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 12A 12B 12C 15A 15B 15C 15D 20A ··· 20H 20I ··· 20T 24A 24B 24C 24D 30A 30B 30C 30D 30E 30F 30G 30H 40A ··· 40H 60A ··· 60H 60I 60J 60K 60L 120A ··· 120P order 1 2 2 2 3 4 4 4 4 4 5 5 5 5 6 6 8 8 10 10 10 10 10 10 10 10 10 10 10 10 12 12 12 15 15 15 15 20 ··· 20 20 ··· 20 24 24 24 24 30 30 30 30 30 30 30 30 40 ··· 40 60 ··· 60 60 60 60 60 120 ··· 120 size 1 1 2 12 2 2 2 12 12 12 1 1 1 1 2 4 4 4 1 1 1 1 2 2 2 2 12 12 12 12 2 2 4 2 2 2 2 2 ··· 2 12 ··· 12 4 4 4 4 2 2 2 2 4 4 4 4 4 ··· 4 2 ··· 2 4 4 4 4 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 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 D12 D12 C5×S3 C5×D4 C5×D4 S3×C10 S3×C10 C5×D12 C5×D12 C8.C22 C8.D6 C5×C8.C22 C5×C8.D6 kernel C5×C8.D6 C5×C24⋊C2 C5×Dic12 C15×M4(2) C10×Dic6 C5×C4○D12 C8.D6 C24⋊C2 Dic12 C3×M4(2) C2×Dic6 C4○D12 C5×M4(2) C60 C2×C30 C40 C2×C20 C20 C2×C10 M4(2) C12 C2×C6 C8 C2×C4 C4 C22 C15 C5 C3 C1 # reps 1 2 2 1 1 1 4 8 8 4 4 4 1 1 1 2 1 2 2 4 4 4 8 4 8 8 1 2 4 8

Matrix representation of C5×C8.D6 in GL4(𝔽241) generated by

 98 0 0 0 0 98 0 0 0 0 98 0 0 0 0 98
,
 0 0 239 0 0 0 0 239 71 142 0 0 99 170 0 0
,
 198 99 0 0 142 99 0 0 0 0 43 142 0 0 99 142
,
 0 0 163 90 0 0 168 78 101 143 0 0 42 140 0 0
G:=sub<GL(4,GF(241))| [98,0,0,0,0,98,0,0,0,0,98,0,0,0,0,98],[0,0,71,99,0,0,142,170,239,0,0,0,0,239,0,0],[198,142,0,0,99,99,0,0,0,0,43,99,0,0,142,142],[0,0,101,42,0,0,143,140,163,168,0,0,90,78,0,0] >;

C5×C8.D6 in GAP, Magma, Sage, TeX

C_5\times C_8.D_6
% in TeX

G:=Group("C5xC8.D6");
// GroupNames label

G:=SmallGroup(480,788);
// by ID

G=gap.SmallGroup(480,788);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,568,926,891,226,4204,102,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=1,c^6=d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

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