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G = C5×C6.D8order 480 = 25·3·5

Direct product of C5 and C6.D8

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

Aliases: C5×C6.D8, D123C20, C30.52D8, C20.61D12, C60.134D4, C30.38SD16, C6.7(C5×D8), C4.1(S3×C20), C4.9(C5×D12), C12.1(C5×D4), (C5×D12)⋊15C4, C20.76(C4×S3), C12.3(C2×C20), C6.7(C5×SD16), C60.173(C2×C4), (C2×D12).5C10, (C2×C20).346D6, (C2×C30).174D4, C1513(D4⋊C4), C10.50(D6⋊C4), C10.23(D4⋊S3), (C10×D12).15C2, C30.92(C22⋊C4), (C2×C60).339C22, C10.11(Q82S3), (C2×C3⋊C8)⋊1C10, C4⋊C41(C5×S3), (C3×C4⋊C4)⋊1C10, (C10×C3⋊C8)⋊15C2, (C5×C4⋊C4)⋊10S3, C31(C5×D4⋊C4), (C15×C4⋊C4)⋊19C2, C2.5(C5×D6⋊C4), C2.2(C5×D4⋊S3), (C2×C6).30(C5×D4), C6.3(C5×C22⋊C4), (C2×C12).9(C2×C10), (C2×C4).34(S3×C10), C2.2(C5×Q82S3), C22.14(C5×C3⋊D4), (C2×C10).86(C3⋊D4), SmallGroup(480,128)

Series: Derived Chief Lower central Upper central

C1C12 — C5×C6.D8
C1C3C6C12C2×C12C2×C60C10×D12 — C5×C6.D8
C3C6C12 — C5×C6.D8
C1C2×C10C2×C20C5×C4⋊C4

Generators and relations for C5×C6.D8
 G = < a,b,c,d | a5=b6=c8=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=b3c-1 >

Subgroups: 308 in 100 conjugacy classes, 46 normal (42 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4, C22, C22 [×4], C5, S3 [×2], C6 [×3], C8, C2×C4, C2×C4, D4 [×3], C23, C10 [×3], C10 [×2], C12 [×2], C12, D6 [×4], C2×C6, C15, C4⋊C4, C2×C8, C2×D4, C20 [×2], C20, C2×C10, C2×C10 [×4], C3⋊C8, D12 [×2], D12, C2×C12, C2×C12, C22×S3, C5×S3 [×2], C30 [×3], D4⋊C4, C40, C2×C20, C2×C20, C5×D4 [×3], C22×C10, C2×C3⋊C8, C3×C4⋊C4, C2×D12, C60 [×2], C60, S3×C10 [×4], C2×C30, C5×C4⋊C4, C2×C40, D4×C10, C6.D8, C5×C3⋊C8, C5×D12 [×2], C5×D12, C2×C60, C2×C60, S3×C2×C10, C5×D4⋊C4, C10×C3⋊C8, C15×C4⋊C4, C10×D12, C5×C6.D8
Quotients: C1, C2 [×3], C4 [×2], C22, C5, S3, C2×C4, D4 [×2], C10 [×3], D6, C22⋊C4, D8, SD16, C20 [×2], C2×C10, C4×S3, D12, C3⋊D4, C5×S3, D4⋊C4, C2×C20, C5×D4 [×2], D6⋊C4, D4⋊S3, Q82S3, S3×C10, C5×C22⋊C4, C5×D8, C5×SD16, C6.D8, S3×C20, C5×D12, C5×C3⋊D4, C5×D4⋊C4, C5×D6⋊C4, C5×D4⋊S3, C5×Q82S3, C5×C6.D8

Smallest permutation representation of C5×C6.D8
On 240 points
Generators in S240
(1 52 183 28 159)(2 53 184 29 160)(3 54 177 30 153)(4 55 178 31 154)(5 56 179 32 155)(6 49 180 25 156)(7 50 181 26 157)(8 51 182 27 158)(9 171 63 147 39)(10 172 64 148 40)(11 173 57 149 33)(12 174 58 150 34)(13 175 59 151 35)(14 176 60 152 36)(15 169 61 145 37)(16 170 62 146 38)(17 99 230 75 206)(18 100 231 76 207)(19 101 232 77 208)(20 102 225 78 201)(21 103 226 79 202)(22 104 227 80 203)(23 97 228 73 204)(24 98 229 74 205)(41 130 186 65 162)(42 131 187 66 163)(43 132 188 67 164)(44 133 189 68 165)(45 134 190 69 166)(46 135 191 70 167)(47 136 192 71 168)(48 129 185 72 161)(81 141 221 105 197)(82 142 222 106 198)(83 143 223 107 199)(84 144 224 108 200)(85 137 217 109 193)(86 138 218 110 194)(87 139 219 111 195)(88 140 220 112 196)(89 123 234 113 210)(90 124 235 114 211)(91 125 236 115 212)(92 126 237 116 213)(93 127 238 117 214)(94 128 239 118 215)(95 121 240 119 216)(96 122 233 120 209)
(1 83 92 203 130 10)(2 11 131 204 93 84)(3 85 94 205 132 12)(4 13 133 206 95 86)(5 87 96 207 134 14)(6 15 135 208 89 88)(7 81 90 201 136 16)(8 9 129 202 91 82)(17 121 138 55 175 189)(18 190 176 56 139 122)(19 123 140 49 169 191)(20 192 170 50 141 124)(21 125 142 51 171 185)(22 186 172 52 143 126)(23 127 144 53 173 187)(24 188 174 54 137 128)(25 145 167 232 113 112)(26 105 114 225 168 146)(27 147 161 226 115 106)(28 107 116 227 162 148)(29 149 163 228 117 108)(30 109 118 229 164 150)(31 151 165 230 119 110)(32 111 120 231 166 152)(33 42 73 214 200 160)(34 153 193 215 74 43)(35 44 75 216 194 154)(36 155 195 209 76 45)(37 46 77 210 196 156)(38 157 197 211 78 47)(39 48 79 212 198 158)(40 159 199 213 80 41)(57 66 97 238 224 184)(58 177 217 239 98 67)(59 68 99 240 218 178)(60 179 219 233 100 69)(61 70 101 234 220 180)(62 181 221 235 102 71)(63 72 103 236 222 182)(64 183 223 237 104 65)
(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)
(1 203)(2 8)(3 201)(4 6)(5 207)(7 205)(9 84)(10 130)(11 82)(12 136)(13 88)(14 134)(15 86)(16 132)(17 19)(18 56)(20 54)(21 23)(22 52)(24 50)(25 31)(26 229)(27 29)(28 227)(30 225)(32 231)(33 198)(34 47)(35 196)(36 45)(37 194)(38 43)(39 200)(40 41)(42 212)(44 210)(46 216)(48 214)(49 55)(51 53)(57 222)(58 71)(59 220)(60 69)(61 218)(62 67)(63 224)(64 65)(66 236)(68 234)(70 240)(72 238)(73 79)(74 157)(75 77)(76 155)(78 153)(80 159)(81 94)(83 92)(85 90)(87 96)(89 133)(91 131)(93 129)(95 135)(97 103)(98 181)(99 101)(100 179)(102 177)(104 183)(105 118)(106 149)(107 116)(108 147)(109 114)(110 145)(111 120)(112 151)(113 165)(115 163)(117 161)(119 167)(121 191)(122 139)(123 189)(124 137)(125 187)(126 143)(127 185)(128 141)(138 169)(140 175)(142 173)(144 171)(146 164)(148 162)(150 168)(152 166)(154 156)(158 160)(170 188)(172 186)(174 192)(176 190)(178 180)(182 184)(193 211)(195 209)(197 215)(199 213)(202 204)(206 208)(217 235)(219 233)(221 239)(223 237)(226 228)(230 232)

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

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

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

120 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D5A5B5C5D6A6B6C8A8B8C8D10A···10L10M···10T12A···12F15A15B15C15D20A···20H20I···20P30A···30L40A···40P60A···60X
order122222344445555666888810···1010···1012···121515151520···2020···2030···3040···4060···60
size1111121222244111122266661···112···124···422222···24···42···26···64···4

120 irreducible representations

dim11111111112222222222222222224444
type++++++++++++
imageC1C2C2C2C4C5C10C10C10C20S3D4D4D6D8SD16C4×S3D12C3⋊D4C5×S3C5×D4C5×D4S3×C10C5×D8C5×SD16S3×C20C5×D12C5×C3⋊D4D4⋊S3Q82S3C5×D4⋊S3C5×Q82S3
kernelC5×C6.D8C10×C3⋊C8C15×C4⋊C4C10×D12C5×D12C6.D8C2×C3⋊C8C3×C4⋊C4C2×D12D12C5×C4⋊C4C60C2×C30C2×C20C30C30C20C20C2×C10C4⋊C4C12C2×C6C2×C4C6C6C4C4C22C10C10C2C2
# reps111144444161111222224444888881144

Matrix representation of C5×C6.D8 in GL4(𝔽241) generated by

91000
09100
0010
0001
,
024000
1100
0010
0001
,
017700
177000
000219
0011219
,
024000
240000
0010
001240
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,240,1,0,0,0,0,1,0,0,0,0,1],[0,177,0,0,177,0,0,0,0,0,0,11,0,0,219,219],[0,240,0,0,240,0,0,0,0,0,1,1,0,0,0,240] >;

C5×C6.D8 in GAP, Magma, Sage, TeX

C_5\times C_6.D_8
% in TeX

G:=Group("C5xC6.D8");
// GroupNames label

G:=SmallGroup(480,128);
// by ID

G=gap.SmallGroup(480,128);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-3,589,148,4204,2111,102,15686]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^6=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=b^3*c^-1>;
// generators/relations

׿
×
𝔽