metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊C8.6D6, D15⋊C8⋊4C2, C15⋊Q8.1C4, C15⋊5(C8○D4), C5⋊3(C8○D12), D6.F5⋊4C2, C3⋊3(D4.F5), D6.3(C2×F5), C3⋊D4.2F5, D30.3(C2×C4), C5⋊D12.1C4, C15⋊7D4.1C4, C22.1(S3×F5), Dic3.F5⋊4C2, Dic5.3(C4×S3), Dic3.5(C2×F5), C6.22(C22×F5), C15⋊8M4(2)⋊2C2, C30.22(C22×C4), C15⋊C8.3C22, Dic15.5(C2×C4), (C2×Dic5).75D6, Dic3.D10.1C2, D30.C2.7C22, (S3×Dic5).7C22, (C3×Dic5).32C23, Dic5.34(C22×S3), (C6×Dic5).143C22, (C6×C5⋊C8)⋊4C2, (C2×C5⋊C8)⋊3S3, (S3×C5⋊C8)⋊4C2, C2.23(C2×S3×F5), C10.22(S3×C2×C4), (C2×C10).1(C4×S3), (C5×C3⋊D4).1C4, (C3×C5⋊C8).6C22, (C2×C6).22(C2×F5), (S3×C10).3(C2×C4), (C2×C30).17(C2×C4), (C5×Dic3).5(C2×C4), (C3×Dic5).24(C2×C4), SmallGroup(480,1003)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊C8.D6
G = < a,b,c,d | a5=b8=1, c6=b6, d2=b2, bab-1=cac-1=dad-1=a3, bc=cb, dbd-1=b5, dcd-1=b4c5 >
Subgroups: 564 in 124 conjugacy classes, 46 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, D4, Q8, D5, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C15, C2×C8, M4(2), C4○D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C3⋊D4, C2×C12, C5×S3, D15, C30, C30, C8○D4, C5⋊C8, C5⋊C8, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C5×D4, S3×C8, C8⋊S3, C4.Dic3, C2×C24, C4○D12, C5×Dic3, C3×Dic5, Dic15, S3×C10, D30, C2×C30, D5⋊C8, C4.F5, C2×C5⋊C8, C2×C5⋊C8, C22.F5, D4⋊2D5, C8○D12, C3×C5⋊C8, C15⋊C8, S3×Dic5, D30.C2, C5⋊D12, C15⋊Q8, C6×Dic5, C5×C3⋊D4, C15⋊7D4, D4.F5, S3×C5⋊C8, D15⋊C8, D6.F5, Dic3.F5, C6×C5⋊C8, C15⋊8M4(2), Dic3.D10, C5⋊C8.D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, F5, C4×S3, C22×S3, C8○D4, C2×F5, S3×C2×C4, C22×F5, C8○D12, S3×F5, D4.F5, C2×S3×F5, C5⋊C8.D6
►On 240 points
Generators in S240
(1 229 35 173 107)(2 174 230 108 36)(3 109 175 37 231)(4 38 110 232 176)(5 233 39 177 111)(6 178 234 112 40)(7 113 179 41 235)(8 42 114 236 180)(9 237 43 181 115)(10 182 238 116 44)(11 117 183 45 239)(12 46 118 240 184)(13 217 47 185 119)(14 186 218 120 48)(15 97 187 25 219)(16 26 98 220 188)(17 221 27 189 99)(18 190 222 100 28)(19 101 191 29 223)(20 30 102 224 192)(21 225 31 169 103)(22 170 226 104 32)(23 105 171 33 227)(24 34 106 228 172)(49 162 73 196 134)(50 197 163 135 74)(51 136 198 75 164)(52 76 137 165 199)(53 166 77 200 138)(54 201 167 139 78)(55 140 202 79 168)(56 80 141 145 203)(57 146 81 204 142)(58 205 147 143 82)(59 144 206 83 148)(60 84 121 149 207)(61 150 85 208 122)(62 209 151 123 86)(63 124 210 87 152)(64 88 125 153 211)(65 154 89 212 126)(66 213 155 127 90)(67 128 214 91 156)(68 92 129 157 215)(69 158 93 216 130)(70 193 159 131 94)(71 132 194 95 160)(72 96 133 161 195)
(1 152 19 146 13 164 7 158)(2 153 20 147 14 165 8 159)(3 154 21 148 15 166 9 160)(4 155 22 149 16 167 10 161)(5 156 23 150 17 168 11 162)(6 157 24 151 18 145 12 163)(25 77 43 95 37 89 31 83)(26 78 44 96 38 90 32 84)(27 79 45 73 39 91 33 85)(28 80 46 74 40 92 34 86)(29 81 47 75 41 93 35 87)(30 82 48 76 42 94 36 88)(49 177 67 171 61 189 55 183)(50 178 68 172 62 190 56 184)(51 179 69 173 63 191 57 185)(52 180 70 174 64 192 58 186)(53 181 71 175 65 169 59 187)(54 182 72 176 66 170 60 188)(97 200 115 194 109 212 103 206)(98 201 116 195 110 213 104 207)(99 202 117 196 111 214 105 208)(100 203 118 197 112 215 106 209)(101 204 119 198 113 216 107 210)(102 205 120 199 114 193 108 211)(121 220 139 238 133 232 127 226)(122 221 140 239 134 233 128 227)(123 222 141 240 135 234 129 228)(124 223 142 217 136 235 130 229)(125 224 143 218 137 236 131 230)(126 225 144 219 138 237 132 231)
(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 18 19 12 13 6 7 24)(2 11 20 5 14 23 8 17)(3 4 21 22 15 16 9 10)(25 26 43 44 37 38 31 32)(27 36 45 30 39 48 33 42)(28 29 46 47 40 41 34 35)(49 70 67 64 61 58 55 52)(50 63 68 57 62 51 56 69)(53 66 71 60 65 54 59 72)(73 94 91 88 85 82 79 76)(74 87 92 81 86 75 80 93)(77 90 95 84 89 78 83 96)(97 98 115 116 109 110 103 104)(99 108 117 102 111 120 105 114)(100 101 118 119 112 113 106 107)(121 126 139 144 133 138 127 132)(122 143 140 137 134 131 128 125)(123 136 141 130 135 124 129 142)(145 158 163 152 157 146 151 164)(147 168 165 162 159 156 153 150)(148 161 166 155 160 149 154 167)(169 170 187 188 181 182 175 176)(171 180 189 174 183 192 177 186)(172 173 190 191 184 185 178 179)(193 214 211 208 205 202 199 196)(194 207 212 201 206 195 200 213)(197 210 215 204 209 198 203 216)(217 234 235 228 229 222 223 240)(218 227 236 221 230 239 224 233)(219 220 237 238 231 232 225 226)
G:=sub<Sym(240)| (1,229,35,173,107)(2,174,230,108,36)(3,109,175,37,231)(4,38,110,232,176)(5,233,39,177,111)(6,178,234,112,40)(7,113,179,41,235)(8,42,114,236,180)(9,237,43,181,115)(10,182,238,116,44)(11,117,183,45,239)(12,46,118,240,184)(13,217,47,185,119)(14,186,218,120,48)(15,97,187,25,219)(16,26,98,220,188)(17,221,27,189,99)(18,190,222,100,28)(19,101,191,29,223)(20,30,102,224,192)(21,225,31,169,103)(22,170,226,104,32)(23,105,171,33,227)(24,34,106,228,172)(49,162,73,196,134)(50,197,163,135,74)(51,136,198,75,164)(52,76,137,165,199)(53,166,77,200,138)(54,201,167,139,78)(55,140,202,79,168)(56,80,141,145,203)(57,146,81,204,142)(58,205,147,143,82)(59,144,206,83,148)(60,84,121,149,207)(61,150,85,208,122)(62,209,151,123,86)(63,124,210,87,152)(64,88,125,153,211)(65,154,89,212,126)(66,213,155,127,90)(67,128,214,91,156)(68,92,129,157,215)(69,158,93,216,130)(70,193,159,131,94)(71,132,194,95,160)(72,96,133,161,195), (1,152,19,146,13,164,7,158)(2,153,20,147,14,165,8,159)(3,154,21,148,15,166,9,160)(4,155,22,149,16,167,10,161)(5,156,23,150,17,168,11,162)(6,157,24,151,18,145,12,163)(25,77,43,95,37,89,31,83)(26,78,44,96,38,90,32,84)(27,79,45,73,39,91,33,85)(28,80,46,74,40,92,34,86)(29,81,47,75,41,93,35,87)(30,82,48,76,42,94,36,88)(49,177,67,171,61,189,55,183)(50,178,68,172,62,190,56,184)(51,179,69,173,63,191,57,185)(52,180,70,174,64,192,58,186)(53,181,71,175,65,169,59,187)(54,182,72,176,66,170,60,188)(97,200,115,194,109,212,103,206)(98,201,116,195,110,213,104,207)(99,202,117,196,111,214,105,208)(100,203,118,197,112,215,106,209)(101,204,119,198,113,216,107,210)(102,205,120,199,114,193,108,211)(121,220,139,238,133,232,127,226)(122,221,140,239,134,233,128,227)(123,222,141,240,135,234,129,228)(124,223,142,217,136,235,130,229)(125,224,143,218,137,236,131,230)(126,225,144,219,138,237,132,231), (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), (1,18,19,12,13,6,7,24)(2,11,20,5,14,23,8,17)(3,4,21,22,15,16,9,10)(25,26,43,44,37,38,31,32)(27,36,45,30,39,48,33,42)(28,29,46,47,40,41,34,35)(49,70,67,64,61,58,55,52)(50,63,68,57,62,51,56,69)(53,66,71,60,65,54,59,72)(73,94,91,88,85,82,79,76)(74,87,92,81,86,75,80,93)(77,90,95,84,89,78,83,96)(97,98,115,116,109,110,103,104)(99,108,117,102,111,120,105,114)(100,101,118,119,112,113,106,107)(121,126,139,144,133,138,127,132)(122,143,140,137,134,131,128,125)(123,136,141,130,135,124,129,142)(145,158,163,152,157,146,151,164)(147,168,165,162,159,156,153,150)(148,161,166,155,160,149,154,167)(169,170,187,188,181,182,175,176)(171,180,189,174,183,192,177,186)(172,173,190,191,184,185,178,179)(193,214,211,208,205,202,199,196)(194,207,212,201,206,195,200,213)(197,210,215,204,209,198,203,216)(217,234,235,228,229,222,223,240)(218,227,236,221,230,239,224,233)(219,220,237,238,231,232,225,226)>;
G:=Group( (1,229,35,173,107)(2,174,230,108,36)(3,109,175,37,231)(4,38,110,232,176)(5,233,39,177,111)(6,178,234,112,40)(7,113,179,41,235)(8,42,114,236,180)(9,237,43,181,115)(10,182,238,116,44)(11,117,183,45,239)(12,46,118,240,184)(13,217,47,185,119)(14,186,218,120,48)(15,97,187,25,219)(16,26,98,220,188)(17,221,27,189,99)(18,190,222,100,28)(19,101,191,29,223)(20,30,102,224,192)(21,225,31,169,103)(22,170,226,104,32)(23,105,171,33,227)(24,34,106,228,172)(49,162,73,196,134)(50,197,163,135,74)(51,136,198,75,164)(52,76,137,165,199)(53,166,77,200,138)(54,201,167,139,78)(55,140,202,79,168)(56,80,141,145,203)(57,146,81,204,142)(58,205,147,143,82)(59,144,206,83,148)(60,84,121,149,207)(61,150,85,208,122)(62,209,151,123,86)(63,124,210,87,152)(64,88,125,153,211)(65,154,89,212,126)(66,213,155,127,90)(67,128,214,91,156)(68,92,129,157,215)(69,158,93,216,130)(70,193,159,131,94)(71,132,194,95,160)(72,96,133,161,195), (1,152,19,146,13,164,7,158)(2,153,20,147,14,165,8,159)(3,154,21,148,15,166,9,160)(4,155,22,149,16,167,10,161)(5,156,23,150,17,168,11,162)(6,157,24,151,18,145,12,163)(25,77,43,95,37,89,31,83)(26,78,44,96,38,90,32,84)(27,79,45,73,39,91,33,85)(28,80,46,74,40,92,34,86)(29,81,47,75,41,93,35,87)(30,82,48,76,42,94,36,88)(49,177,67,171,61,189,55,183)(50,178,68,172,62,190,56,184)(51,179,69,173,63,191,57,185)(52,180,70,174,64,192,58,186)(53,181,71,175,65,169,59,187)(54,182,72,176,66,170,60,188)(97,200,115,194,109,212,103,206)(98,201,116,195,110,213,104,207)(99,202,117,196,111,214,105,208)(100,203,118,197,112,215,106,209)(101,204,119,198,113,216,107,210)(102,205,120,199,114,193,108,211)(121,220,139,238,133,232,127,226)(122,221,140,239,134,233,128,227)(123,222,141,240,135,234,129,228)(124,223,142,217,136,235,130,229)(125,224,143,218,137,236,131,230)(126,225,144,219,138,237,132,231), (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), (1,18,19,12,13,6,7,24)(2,11,20,5,14,23,8,17)(3,4,21,22,15,16,9,10)(25,26,43,44,37,38,31,32)(27,36,45,30,39,48,33,42)(28,29,46,47,40,41,34,35)(49,70,67,64,61,58,55,52)(50,63,68,57,62,51,56,69)(53,66,71,60,65,54,59,72)(73,94,91,88,85,82,79,76)(74,87,92,81,86,75,80,93)(77,90,95,84,89,78,83,96)(97,98,115,116,109,110,103,104)(99,108,117,102,111,120,105,114)(100,101,118,119,112,113,106,107)(121,126,139,144,133,138,127,132)(122,143,140,137,134,131,128,125)(123,136,141,130,135,124,129,142)(145,158,163,152,157,146,151,164)(147,168,165,162,159,156,153,150)(148,161,166,155,160,149,154,167)(169,170,187,188,181,182,175,176)(171,180,189,174,183,192,177,186)(172,173,190,191,184,185,178,179)(193,214,211,208,205,202,199,196)(194,207,212,201,206,195,200,213)(197,210,215,204,209,198,203,216)(217,234,235,228,229,222,223,240)(218,227,236,221,230,239,224,233)(219,220,237,238,231,232,225,226) );
G=PermutationGroup([[(1,229,35,173,107),(2,174,230,108,36),(3,109,175,37,231),(4,38,110,232,176),(5,233,39,177,111),(6,178,234,112,40),(7,113,179,41,235),(8,42,114,236,180),(9,237,43,181,115),(10,182,238,116,44),(11,117,183,45,239),(12,46,118,240,184),(13,217,47,185,119),(14,186,218,120,48),(15,97,187,25,219),(16,26,98,220,188),(17,221,27,189,99),(18,190,222,100,28),(19,101,191,29,223),(20,30,102,224,192),(21,225,31,169,103),(22,170,226,104,32),(23,105,171,33,227),(24,34,106,228,172),(49,162,73,196,134),(50,197,163,135,74),(51,136,198,75,164),(52,76,137,165,199),(53,166,77,200,138),(54,201,167,139,78),(55,140,202,79,168),(56,80,141,145,203),(57,146,81,204,142),(58,205,147,143,82),(59,144,206,83,148),(60,84,121,149,207),(61,150,85,208,122),(62,209,151,123,86),(63,124,210,87,152),(64,88,125,153,211),(65,154,89,212,126),(66,213,155,127,90),(67,128,214,91,156),(68,92,129,157,215),(69,158,93,216,130),(70,193,159,131,94),(71,132,194,95,160),(72,96,133,161,195)], [(1,152,19,146,13,164,7,158),(2,153,20,147,14,165,8,159),(3,154,21,148,15,166,9,160),(4,155,22,149,16,167,10,161),(5,156,23,150,17,168,11,162),(6,157,24,151,18,145,12,163),(25,77,43,95,37,89,31,83),(26,78,44,96,38,90,32,84),(27,79,45,73,39,91,33,85),(28,80,46,74,40,92,34,86),(29,81,47,75,41,93,35,87),(30,82,48,76,42,94,36,88),(49,177,67,171,61,189,55,183),(50,178,68,172,62,190,56,184),(51,179,69,173,63,191,57,185),(52,180,70,174,64,192,58,186),(53,181,71,175,65,169,59,187),(54,182,72,176,66,170,60,188),(97,200,115,194,109,212,103,206),(98,201,116,195,110,213,104,207),(99,202,117,196,111,214,105,208),(100,203,118,197,112,215,106,209),(101,204,119,198,113,216,107,210),(102,205,120,199,114,193,108,211),(121,220,139,238,133,232,127,226),(122,221,140,239,134,233,128,227),(123,222,141,240,135,234,129,228),(124,223,142,217,136,235,130,229),(125,224,143,218,137,236,131,230),(126,225,144,219,138,237,132,231)], [(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)], [(1,18,19,12,13,6,7,24),(2,11,20,5,14,23,8,17),(3,4,21,22,15,16,9,10),(25,26,43,44,37,38,31,32),(27,36,45,30,39,48,33,42),(28,29,46,47,40,41,34,35),(49,70,67,64,61,58,55,52),(50,63,68,57,62,51,56,69),(53,66,71,60,65,54,59,72),(73,94,91,88,85,82,79,76),(74,87,92,81,86,75,80,93),(77,90,95,84,89,78,83,96),(97,98,115,116,109,110,103,104),(99,108,117,102,111,120,105,114),(100,101,118,119,112,113,106,107),(121,126,139,144,133,138,127,132),(122,143,140,137,134,131,128,125),(123,136,141,130,135,124,129,142),(145,158,163,152,157,146,151,164),(147,168,165,162,159,156,153,150),(148,161,166,155,160,149,154,167),(169,170,187,188,181,182,175,176),(171,180,189,174,183,192,177,186),(172,173,190,191,184,185,178,179),(193,214,211,208,205,202,199,196),(194,207,212,201,206,195,200,213),(197,210,215,204,209,198,203,216),(217,234,235,228,229,222,223,240),(218,227,236,221,230,239,224,233),(219,220,237,238,231,232,225,226)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 15 | 20 | 24A | ··· | 24H | 30A | 30B | 30C |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 20 | 24 | ··· | 24 | 30 | 30 | 30 |
| size | 1 | 1 | 2 | 6 | 30 | 2 | 5 | 5 | 6 | 10 | 30 | 4 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 30 | 30 | 30 | 30 | 4 | 8 | 24 | 10 | 10 | 10 | 10 | 8 | 24 | 10 | ··· | 10 | 8 | 8 | 8 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D6 | D6 | C4×S3 | C4×S3 | C8○D4 | C8○D12 | F5 | C2×F5 | C2×F5 | C2×F5 | S3×F5 | D4.F5 | C2×S3×F5 | C5⋊C8.D6 |
| kernel | C5⋊C8.D6 | S3×C5⋊C8 | D15⋊C8 | D6.F5 | Dic3.F5 | C6×C5⋊C8 | C15⋊8M4(2) | Dic3.D10 | C5⋊D12 | C15⋊Q8 | C5×C3⋊D4 | C15⋊7D4 | C2×C5⋊C8 | C5⋊C8 | C2×Dic5 | Dic5 | C2×C10 | C15 | C5 | C3⋊D4 | Dic3 | D6 | C2×C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 |
Matrix representation of C5⋊C8.D6 ►in GL8(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 240 |
| 30 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 60 | 211 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 45 | 160 | 165 | 55 |
| 0 | 0 | 0 | 0 | 210 | 215 | 222 | 100 |
| 0 | 0 | 0 | 0 | 26 | 19 | 141 | 24 |
| 0 | 0 | 0 | 0 | 186 | 184 | 196 | 81 |
| 211 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 211 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 45 | 160 | 165 | 55 |
| 0 | 0 | 0 | 0 | 210 | 215 | 222 | 100 |
| 0 | 0 | 0 | 0 | 26 | 19 | 141 | 24 |
| 0 | 0 | 0 | 0 | 186 | 184 | 196 | 81 |
| 211 | 30 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 196 | 81 | 76 | 186 |
| 0 | 0 | 0 | 0 | 31 | 26 | 19 | 141 |
| 0 | 0 | 0 | 0 | 215 | 222 | 100 | 217 |
| 0 | 0 | 0 | 0 | 55 | 57 | 45 | 160 |
G:=sub<GL(8,GF(241))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240,240,240],[30,60,0,0,0,0,0,0,0,211,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,45,210,26,186,0,0,0,0,160,215,19,184,0,0,0,0,165,222,141,196,0,0,0,0,55,100,24,81],[211,0,0,0,0,0,0,0,0,211,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,45,210,26,186,0,0,0,0,160,215,19,184,0,0,0,0,165,222,141,196,0,0,0,0,55,100,24,81],[211,0,0,0,0,0,0,0,30,30,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,196,31,215,55,0,0,0,0,81,26,222,57,0,0,0,0,76,19,100,45,0,0,0,0,186,141,217,160] >;
C5⋊C8.D6 in GAP, Magma, Sage, TeX
C_5\rtimes C_8.D_6
% in TeX
G:=Group("C5:C8.D6"); // GroupNames label
G:=SmallGroup(480,1003);
// by ID
G=gap.SmallGroup(480,1003);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,219,80,1356,9414,2379]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=1,c^6=b^6,d^2=b^2,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^3,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=b^4*c^5>;
// generators/relations