direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5×C6.D8, D12⋊3C20, 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, C15⋊13(D4⋊C4), C10.50(D6⋊C4), C10.23(D4⋊S3), (C10×D12).15C2, C30.92(C22⋊C4), (C2×C60).339C22, C10.11(Q8⋊2S3), (C2×C3⋊C8)⋊1C10, C4⋊C4⋊1(C5×S3), (C3×C4⋊C4)⋊1C10, (C10×C3⋊C8)⋊15C2, (C5×C4⋊C4)⋊10S3, C3⋊1(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×Q8⋊2S3), C22.14(C5×C3⋊D4), (C2×C10).86(C3⋊D4), SmallGroup(480,128)
Series: Derived ►Chief ►Lower central ►Upper central
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, C2, C3, C4, C4, C22, C22, C5, S3, C6, C8, C2×C4, C2×C4, D4, C23, C10, C10, C12, C12, D6, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, C20, C20, C2×C10, C2×C10, C3⋊C8, D12, D12, C2×C12, C2×C12, C22×S3, C5×S3, C30, D4⋊C4, C40, C2×C20, C2×C20, C5×D4, C22×C10, C2×C3⋊C8, C3×C4⋊C4, C2×D12, C60, C60, S3×C10, C2×C30, C5×C4⋊C4, C2×C40, D4×C10, C6.D8, C5×C3⋊C8, C5×D12, 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, C4, C22, C5, S3, C2×C4, D4, C10, D6, C22⋊C4, D8, SD16, C20, C2×C10, C4×S3, D12, C3⋊D4, C5×S3, D4⋊C4, C2×C20, C5×D4, D6⋊C4, D4⋊S3, Q8⋊2S3, 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×Q8⋊2S3, C5×C6.D8
►On 240 points
Generators in S240
(1 174 58 150 34)(2 175 59 151 35)(3 176 60 152 36)(4 169 61 145 37)(5 170 62 146 38)(6 171 63 147 39)(7 172 64 148 40)(8 173 57 149 33)(9 52 183 28 159)(10 53 184 29 160)(11 54 177 30 153)(12 55 178 31 154)(13 56 179 32 155)(14 49 180 25 156)(15 50 181 26 157)(16 51 182 27 158)(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 12)(2 13 131 204 93 84)(3 85 94 205 132 14)(4 15 133 206 95 86)(5 87 96 207 134 16)(6 9 135 208 89 88)(7 81 90 201 136 10)(8 11 129 202 91 82)(17 121 138 169 50 189)(18 190 51 170 139 122)(19 123 140 171 52 191)(20 192 53 172 141 124)(21 125 142 173 54 185)(22 186 55 174 143 126)(23 127 144 175 56 187)(24 188 49 176 137 128)(25 152 109 118 229 164)(26 165 230 119 110 145)(27 146 111 120 231 166)(28 167 232 113 112 147)(29 148 105 114 225 168)(30 161 226 115 106 149)(31 150 107 116 227 162)(32 163 228 117 108 151)(33 153 48 79 212 198)(34 199 213 80 41 154)(35 155 42 73 214 200)(36 193 215 74 43 156)(37 157 44 75 216 194)(38 195 209 76 45 158)(39 159 46 77 210 196)(40 197 211 78 47 160)(57 177 72 103 236 222)(58 223 237 104 65 178)(59 179 66 97 238 224)(60 217 239 98 67 180)(61 181 68 99 240 218)(62 219 233 100 69 182)(63 183 70 101 234 220)(64 221 235 102 71 184)
(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 86)(10 132)(11 84)(12 130)(13 82)(14 136)(15 88)(16 134)(17 19)(18 170)(20 176)(21 23)(22 174)(24 172)(25 168)(26 112)(27 166)(28 110)(29 164)(30 108)(31 162)(32 106)(33 35)(34 80)(36 78)(37 39)(38 76)(40 74)(41 154)(42 212)(43 160)(44 210)(45 158)(46 216)(47 156)(48 214)(49 192)(50 140)(51 190)(52 138)(53 188)(54 144)(55 186)(56 142)(57 59)(58 104)(60 102)(61 63)(62 100)(64 98)(65 178)(66 236)(67 184)(68 234)(69 182)(70 240)(71 180)(72 238)(73 79)(75 77)(81 94)(83 92)(85 90)(87 96)(89 133)(91 131)(93 129)(95 135)(97 103)(99 101)(105 118)(107 116)(109 114)(111 120)(113 165)(115 163)(117 161)(119 167)(121 191)(122 139)(123 189)(124 137)(125 187)(126 143)(127 185)(128 141)(145 147)(146 231)(148 229)(149 151)(150 227)(152 225)(153 200)(155 198)(157 196)(159 194)(169 171)(173 175)(177 224)(179 222)(181 220)(183 218)(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)| (1,174,58,150,34)(2,175,59,151,35)(3,176,60,152,36)(4,169,61,145,37)(5,170,62,146,38)(6,171,63,147,39)(7,172,64,148,40)(8,173,57,149,33)(9,52,183,28,159)(10,53,184,29,160)(11,54,177,30,153)(12,55,178,31,154)(13,56,179,32,155)(14,49,180,25,156)(15,50,181,26,157)(16,51,182,27,158)(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), (1,83,92,203,130,12)(2,13,131,204,93,84)(3,85,94,205,132,14)(4,15,133,206,95,86)(5,87,96,207,134,16)(6,9,135,208,89,88)(7,81,90,201,136,10)(8,11,129,202,91,82)(17,121,138,169,50,189)(18,190,51,170,139,122)(19,123,140,171,52,191)(20,192,53,172,141,124)(21,125,142,173,54,185)(22,186,55,174,143,126)(23,127,144,175,56,187)(24,188,49,176,137,128)(25,152,109,118,229,164)(26,165,230,119,110,145)(27,146,111,120,231,166)(28,167,232,113,112,147)(29,148,105,114,225,168)(30,161,226,115,106,149)(31,150,107,116,227,162)(32,163,228,117,108,151)(33,153,48,79,212,198)(34,199,213,80,41,154)(35,155,42,73,214,200)(36,193,215,74,43,156)(37,157,44,75,216,194)(38,195,209,76,45,158)(39,159,46,77,210,196)(40,197,211,78,47,160)(57,177,72,103,236,222)(58,223,237,104,65,178)(59,179,66,97,238,224)(60,217,239,98,67,180)(61,181,68,99,240,218)(62,219,233,100,69,182)(63,183,70,101,234,220)(64,221,235,102,71,184), (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)(9,86)(10,132)(11,84)(12,130)(13,82)(14,136)(15,88)(16,134)(17,19)(18,170)(20,176)(21,23)(22,174)(24,172)(25,168)(26,112)(27,166)(28,110)(29,164)(30,108)(31,162)(32,106)(33,35)(34,80)(36,78)(37,39)(38,76)(40,74)(41,154)(42,212)(43,160)(44,210)(45,158)(46,216)(47,156)(48,214)(49,192)(50,140)(51,190)(52,138)(53,188)(54,144)(55,186)(56,142)(57,59)(58,104)(60,102)(61,63)(62,100)(64,98)(65,178)(66,236)(67,184)(68,234)(69,182)(70,240)(71,180)(72,238)(73,79)(75,77)(81,94)(83,92)(85,90)(87,96)(89,133)(91,131)(93,129)(95,135)(97,103)(99,101)(105,118)(107,116)(109,114)(111,120)(113,165)(115,163)(117,161)(119,167)(121,191)(122,139)(123,189)(124,137)(125,187)(126,143)(127,185)(128,141)(145,147)(146,231)(148,229)(149,151)(150,227)(152,225)(153,200)(155,198)(157,196)(159,194)(169,171)(173,175)(177,224)(179,222)(181,220)(183,218)(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( (1,174,58,150,34)(2,175,59,151,35)(3,176,60,152,36)(4,169,61,145,37)(5,170,62,146,38)(6,171,63,147,39)(7,172,64,148,40)(8,173,57,149,33)(9,52,183,28,159)(10,53,184,29,160)(11,54,177,30,153)(12,55,178,31,154)(13,56,179,32,155)(14,49,180,25,156)(15,50,181,26,157)(16,51,182,27,158)(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), (1,83,92,203,130,12)(2,13,131,204,93,84)(3,85,94,205,132,14)(4,15,133,206,95,86)(5,87,96,207,134,16)(6,9,135,208,89,88)(7,81,90,201,136,10)(8,11,129,202,91,82)(17,121,138,169,50,189)(18,190,51,170,139,122)(19,123,140,171,52,191)(20,192,53,172,141,124)(21,125,142,173,54,185)(22,186,55,174,143,126)(23,127,144,175,56,187)(24,188,49,176,137,128)(25,152,109,118,229,164)(26,165,230,119,110,145)(27,146,111,120,231,166)(28,167,232,113,112,147)(29,148,105,114,225,168)(30,161,226,115,106,149)(31,150,107,116,227,162)(32,163,228,117,108,151)(33,153,48,79,212,198)(34,199,213,80,41,154)(35,155,42,73,214,200)(36,193,215,74,43,156)(37,157,44,75,216,194)(38,195,209,76,45,158)(39,159,46,77,210,196)(40,197,211,78,47,160)(57,177,72,103,236,222)(58,223,237,104,65,178)(59,179,66,97,238,224)(60,217,239,98,67,180)(61,181,68,99,240,218)(62,219,233,100,69,182)(63,183,70,101,234,220)(64,221,235,102,71,184), (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)(9,86)(10,132)(11,84)(12,130)(13,82)(14,136)(15,88)(16,134)(17,19)(18,170)(20,176)(21,23)(22,174)(24,172)(25,168)(26,112)(27,166)(28,110)(29,164)(30,108)(31,162)(32,106)(33,35)(34,80)(36,78)(37,39)(38,76)(40,74)(41,154)(42,212)(43,160)(44,210)(45,158)(46,216)(47,156)(48,214)(49,192)(50,140)(51,190)(52,138)(53,188)(54,144)(55,186)(56,142)(57,59)(58,104)(60,102)(61,63)(62,100)(64,98)(65,178)(66,236)(67,184)(68,234)(69,182)(70,240)(71,180)(72,238)(73,79)(75,77)(81,94)(83,92)(85,90)(87,96)(89,133)(91,131)(93,129)(95,135)(97,103)(99,101)(105,118)(107,116)(109,114)(111,120)(113,165)(115,163)(117,161)(119,167)(121,191)(122,139)(123,189)(124,137)(125,187)(126,143)(127,185)(128,141)(145,147)(146,231)(148,229)(149,151)(150,227)(152,225)(153,200)(155,198)(157,196)(159,194)(169,171)(173,175)(177,224)(179,222)(181,220)(183,218)(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([[(1,174,58,150,34),(2,175,59,151,35),(3,176,60,152,36),(4,169,61,145,37),(5,170,62,146,38),(6,171,63,147,39),(7,172,64,148,40),(8,173,57,149,33),(9,52,183,28,159),(10,53,184,29,160),(11,54,177,30,153),(12,55,178,31,154),(13,56,179,32,155),(14,49,180,25,156),(15,50,181,26,157),(16,51,182,27,158),(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)], [(1,83,92,203,130,12),(2,13,131,204,93,84),(3,85,94,205,132,14),(4,15,133,206,95,86),(5,87,96,207,134,16),(6,9,135,208,89,88),(7,81,90,201,136,10),(8,11,129,202,91,82),(17,121,138,169,50,189),(18,190,51,170,139,122),(19,123,140,171,52,191),(20,192,53,172,141,124),(21,125,142,173,54,185),(22,186,55,174,143,126),(23,127,144,175,56,187),(24,188,49,176,137,128),(25,152,109,118,229,164),(26,165,230,119,110,145),(27,146,111,120,231,166),(28,167,232,113,112,147),(29,148,105,114,225,168),(30,161,226,115,106,149),(31,150,107,116,227,162),(32,163,228,117,108,151),(33,153,48,79,212,198),(34,199,213,80,41,154),(35,155,42,73,214,200),(36,193,215,74,43,156),(37,157,44,75,216,194),(38,195,209,76,45,158),(39,159,46,77,210,196),(40,197,211,78,47,160),(57,177,72,103,236,222),(58,223,237,104,65,178),(59,179,66,97,238,224),(60,217,239,98,67,180),(61,181,68,99,240,218),(62,219,233,100,69,182),(63,183,70,101,234,220),(64,221,235,102,71,184)], [(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),(9,86),(10,132),(11,84),(12,130),(13,82),(14,136),(15,88),(16,134),(17,19),(18,170),(20,176),(21,23),(22,174),(24,172),(25,168),(26,112),(27,166),(28,110),(29,164),(30,108),(31,162),(32,106),(33,35),(34,80),(36,78),(37,39),(38,76),(40,74),(41,154),(42,212),(43,160),(44,210),(45,158),(46,216),(47,156),(48,214),(49,192),(50,140),(51,190),(52,138),(53,188),(54,144),(55,186),(56,142),(57,59),(58,104),(60,102),(61,63),(62,100),(64,98),(65,178),(66,236),(67,184),(68,234),(69,182),(70,240),(71,180),(72,238),(73,79),(75,77),(81,94),(83,92),(85,90),(87,96),(89,133),(91,131),(93,129),(95,135),(97,103),(99,101),(105,118),(107,116),(109,114),(111,120),(113,165),(115,163),(117,161),(119,167),(121,191),(122,139),(123,189),(124,137),(125,187),(126,143),(127,185),(128,141),(145,147),(146,231),(148,229),(149,151),(150,227),(152,225),(153,200),(155,198),(157,196),(159,194),(169,171),(173,175),(177,224),(179,222),(181,220),(183,218),(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 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 12 | ··· | 12 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C20 | S3 | D4 | D4 | D6 | D8 | SD16 | C4×S3 | D12 | C3⋊D4 | C5×S3 | C5×D4 | C5×D4 | S3×C10 | C5×D8 | C5×SD16 | S3×C20 | C5×D12 | C5×C3⋊D4 | D4⋊S3 | Q8⋊2S3 | C5×D4⋊S3 | C5×Q8⋊2S3 |
| kernel | C5×C6.D8 | C10×C3⋊C8 | C15×C4⋊C4 | C10×D12 | C5×D12 | C6.D8 | C2×C3⋊C8 | C3×C4⋊C4 | C2×D12 | D12 | C5×C4⋊C4 | C60 | C2×C30 | C2×C20 | C30 | C30 | C20 | C20 | C2×C10 | C4⋊C4 | C12 | C2×C6 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 16 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 1 | 1 | 4 | 4 |
Matrix representation of C5×C6.D8 ►in GL4(𝔽241) generated by
| 91 | 0 | 0 | 0 |
| 0 | 91 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 240 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 177 | 0 | 0 |
| 177 | 0 | 0 | 0 |
| 0 | 0 | 0 | 219 |
| 0 | 0 | 11 | 219 |
| 0 | 240 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 240 |
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