metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊C8.3D6, C3⋊D4.F5, D15⋊C8⋊5C2, C15⋊Q8.2C4, C15⋊6(C8○D4), D6.F5⋊6C2, C3⋊4(D4.F5), D6.4(C2×F5), C5⋊3(D12.C4), D30.4(C2×C4), C15⋊7D4.2C4, C5⋊D12.2C4, C22.F5⋊2S3, C22.2(S3×F5), Dic3.F5⋊5C2, Dic5.4(C4×S3), Dic3.6(C2×F5), C6.24(C22×F5), C30.24(C22×C4), C15⋊C8.6C22, (C2×Dic5).76D6, Dic15.6(C2×C4), Dic3.D10.2C2, D30.C2.8C22, (S3×Dic5).8C22, (C3×Dic5).34C23, Dic5.36(C22×S3), (C6×Dic5).145C22, (S3×C5⋊C8)⋊6C2, C2.25(C2×S3×F5), C10.24(S3×C2×C4), (C2×C15⋊C8)⋊4C2, (C2×C6).7(C2×F5), (C2×C10).2(C4×S3), (C5×C3⋊D4).2C4, (C3×C5⋊C8).3C22, (S3×C10).4(C2×C4), (C2×C30).19(C2×C4), (C3×C22.F5)⋊3C2, (C5×Dic3).6(C2×C4), (C3×Dic5).26(C2×C4), SmallGroup(480,1005)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D15⋊C8⋊C2
G = < a,b,c,d | a15=b4=1, c2=d4=b2, bab-1=a11, cac-1=a4, dad-1=a13, cbc-1=dbd-1=b-1, dcd-1=b2c >
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, C2×C3⋊C8, C3×M4(2), C4○D12, C5×Dic3, C3×Dic5, Dic15, S3×C10, D30, C2×C30, D5⋊C8, C4.F5, C2×C5⋊C8, C22.F5, C22.F5, D4⋊2D5, D12.C4, 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, C3×C22.F5, C2×C15⋊C8, Dic3.D10, D15⋊C8⋊C2
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, D12.C4, S3×F5, D4.F5, C2×S3×F5, D15⋊C8⋊C2
►On 240 points
Generators in S240
(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 76 25 72)(2 87 26 68)(3 83 27 64)(4 79 28 75)(5 90 29 71)(6 86 30 67)(7 82 16 63)(8 78 17 74)(9 89 18 70)(10 85 19 66)(11 81 20 62)(12 77 21 73)(13 88 22 69)(14 84 23 65)(15 80 24 61)(31 110 53 102)(32 106 54 98)(33 117 55 94)(34 113 56 105)(35 109 57 101)(36 120 58 97)(37 116 59 93)(38 112 60 104)(39 108 46 100)(40 119 47 96)(41 115 48 92)(42 111 49 103)(43 107 50 99)(44 118 51 95)(45 114 52 91)(121 187 149 209)(122 183 150 205)(123 194 136 201)(124 190 137 197)(125 186 138 208)(126 182 139 204)(127 193 140 200)(128 189 141 196)(129 185 142 207)(130 181 143 203)(131 192 144 199)(132 188 145 210)(133 184 146 206)(134 195 147 202)(135 191 148 198)(151 211 176 226)(152 222 177 237)(153 218 178 233)(154 214 179 229)(155 225 180 240)(156 221 166 236)(157 217 167 232)(158 213 168 228)(159 224 169 239)(160 220 170 235)(161 216 171 231)(162 212 172 227)(163 223 173 238)(164 219 174 234)(165 215 175 230)
(1 107 25 99)(2 111 26 103)(3 115 27 92)(4 119 28 96)(5 108 29 100)(6 112 30 104)(7 116 16 93)(8 120 17 97)(9 109 18 101)(10 113 19 105)(11 117 20 94)(12 106 21 98)(13 110 22 102)(14 114 23 91)(15 118 24 95)(31 69 53 88)(32 73 54 77)(33 62 55 81)(34 66 56 85)(35 70 57 89)(36 74 58 78)(37 63 59 82)(38 67 60 86)(39 71 46 90)(40 75 47 79)(41 64 48 83)(42 68 49 87)(43 72 50 76)(44 61 51 80)(45 65 52 84)(121 221 149 236)(122 225 150 240)(123 214 136 229)(124 218 137 233)(125 222 138 237)(126 211 139 226)(127 215 140 230)(128 219 141 234)(129 223 142 238)(130 212 143 227)(131 216 144 231)(132 220 145 235)(133 224 146 239)(134 213 147 228)(135 217 148 232)(151 204 176 182)(152 208 177 186)(153 197 178 190)(154 201 179 194)(155 205 180 183)(156 209 166 187)(157 198 167 191)(158 202 168 195)(159 206 169 184)(160 210 170 188)(161 199 171 192)(162 203 172 181)(163 207 173 185)(164 196 174 189)(165 200 175 193)
(1 166 50 149 25 156 43 121)(2 173 54 147 26 163 32 134)(3 180 58 145 27 155 36 132)(4 172 47 143 28 162 40 130)(5 179 51 141 29 154 44 128)(6 171 55 139 30 161 33 126)(7 178 59 137 16 153 37 124)(8 170 48 150 17 160 41 122)(9 177 52 148 18 152 45 135)(10 169 56 146 19 159 34 133)(11 176 60 144 20 151 38 131)(12 168 49 142 21 158 42 129)(13 175 53 140 22 165 31 127)(14 167 57 138 23 157 35 125)(15 174 46 136 24 164 39 123)(61 234 108 201 80 219 100 194)(62 226 112 199 81 211 104 192)(63 233 116 197 82 218 93 190)(64 240 120 210 83 225 97 188)(65 232 109 208 84 217 101 186)(66 239 113 206 85 224 105 184)(67 231 117 204 86 216 94 182)(68 238 106 202 87 223 98 195)(69 230 110 200 88 215 102 193)(70 237 114 198 89 222 91 191)(71 229 118 196 90 214 95 189)(72 236 107 209 76 221 99 187)(73 228 111 207 77 213 103 185)(74 235 115 205 78 220 92 183)(75 227 119 203 79 212 96 181)
G:=sub<Sym(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), (1,76,25,72)(2,87,26,68)(3,83,27,64)(4,79,28,75)(5,90,29,71)(6,86,30,67)(7,82,16,63)(8,78,17,74)(9,89,18,70)(10,85,19,66)(11,81,20,62)(12,77,21,73)(13,88,22,69)(14,84,23,65)(15,80,24,61)(31,110,53,102)(32,106,54,98)(33,117,55,94)(34,113,56,105)(35,109,57,101)(36,120,58,97)(37,116,59,93)(38,112,60,104)(39,108,46,100)(40,119,47,96)(41,115,48,92)(42,111,49,103)(43,107,50,99)(44,118,51,95)(45,114,52,91)(121,187,149,209)(122,183,150,205)(123,194,136,201)(124,190,137,197)(125,186,138,208)(126,182,139,204)(127,193,140,200)(128,189,141,196)(129,185,142,207)(130,181,143,203)(131,192,144,199)(132,188,145,210)(133,184,146,206)(134,195,147,202)(135,191,148,198)(151,211,176,226)(152,222,177,237)(153,218,178,233)(154,214,179,229)(155,225,180,240)(156,221,166,236)(157,217,167,232)(158,213,168,228)(159,224,169,239)(160,220,170,235)(161,216,171,231)(162,212,172,227)(163,223,173,238)(164,219,174,234)(165,215,175,230), (1,107,25,99)(2,111,26,103)(3,115,27,92)(4,119,28,96)(5,108,29,100)(6,112,30,104)(7,116,16,93)(8,120,17,97)(9,109,18,101)(10,113,19,105)(11,117,20,94)(12,106,21,98)(13,110,22,102)(14,114,23,91)(15,118,24,95)(31,69,53,88)(32,73,54,77)(33,62,55,81)(34,66,56,85)(35,70,57,89)(36,74,58,78)(37,63,59,82)(38,67,60,86)(39,71,46,90)(40,75,47,79)(41,64,48,83)(42,68,49,87)(43,72,50,76)(44,61,51,80)(45,65,52,84)(121,221,149,236)(122,225,150,240)(123,214,136,229)(124,218,137,233)(125,222,138,237)(126,211,139,226)(127,215,140,230)(128,219,141,234)(129,223,142,238)(130,212,143,227)(131,216,144,231)(132,220,145,235)(133,224,146,239)(134,213,147,228)(135,217,148,232)(151,204,176,182)(152,208,177,186)(153,197,178,190)(154,201,179,194)(155,205,180,183)(156,209,166,187)(157,198,167,191)(158,202,168,195)(159,206,169,184)(160,210,170,188)(161,199,171,192)(162,203,172,181)(163,207,173,185)(164,196,174,189)(165,200,175,193), (1,166,50,149,25,156,43,121)(2,173,54,147,26,163,32,134)(3,180,58,145,27,155,36,132)(4,172,47,143,28,162,40,130)(5,179,51,141,29,154,44,128)(6,171,55,139,30,161,33,126)(7,178,59,137,16,153,37,124)(8,170,48,150,17,160,41,122)(9,177,52,148,18,152,45,135)(10,169,56,146,19,159,34,133)(11,176,60,144,20,151,38,131)(12,168,49,142,21,158,42,129)(13,175,53,140,22,165,31,127)(14,167,57,138,23,157,35,125)(15,174,46,136,24,164,39,123)(61,234,108,201,80,219,100,194)(62,226,112,199,81,211,104,192)(63,233,116,197,82,218,93,190)(64,240,120,210,83,225,97,188)(65,232,109,208,84,217,101,186)(66,239,113,206,85,224,105,184)(67,231,117,204,86,216,94,182)(68,238,106,202,87,223,98,195)(69,230,110,200,88,215,102,193)(70,237,114,198,89,222,91,191)(71,229,118,196,90,214,95,189)(72,236,107,209,76,221,99,187)(73,228,111,207,77,213,103,185)(74,235,115,205,78,220,92,183)(75,227,119,203,79,212,96,181)>;
G:=Group( (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), (1,76,25,72)(2,87,26,68)(3,83,27,64)(4,79,28,75)(5,90,29,71)(6,86,30,67)(7,82,16,63)(8,78,17,74)(9,89,18,70)(10,85,19,66)(11,81,20,62)(12,77,21,73)(13,88,22,69)(14,84,23,65)(15,80,24,61)(31,110,53,102)(32,106,54,98)(33,117,55,94)(34,113,56,105)(35,109,57,101)(36,120,58,97)(37,116,59,93)(38,112,60,104)(39,108,46,100)(40,119,47,96)(41,115,48,92)(42,111,49,103)(43,107,50,99)(44,118,51,95)(45,114,52,91)(121,187,149,209)(122,183,150,205)(123,194,136,201)(124,190,137,197)(125,186,138,208)(126,182,139,204)(127,193,140,200)(128,189,141,196)(129,185,142,207)(130,181,143,203)(131,192,144,199)(132,188,145,210)(133,184,146,206)(134,195,147,202)(135,191,148,198)(151,211,176,226)(152,222,177,237)(153,218,178,233)(154,214,179,229)(155,225,180,240)(156,221,166,236)(157,217,167,232)(158,213,168,228)(159,224,169,239)(160,220,170,235)(161,216,171,231)(162,212,172,227)(163,223,173,238)(164,219,174,234)(165,215,175,230), (1,107,25,99)(2,111,26,103)(3,115,27,92)(4,119,28,96)(5,108,29,100)(6,112,30,104)(7,116,16,93)(8,120,17,97)(9,109,18,101)(10,113,19,105)(11,117,20,94)(12,106,21,98)(13,110,22,102)(14,114,23,91)(15,118,24,95)(31,69,53,88)(32,73,54,77)(33,62,55,81)(34,66,56,85)(35,70,57,89)(36,74,58,78)(37,63,59,82)(38,67,60,86)(39,71,46,90)(40,75,47,79)(41,64,48,83)(42,68,49,87)(43,72,50,76)(44,61,51,80)(45,65,52,84)(121,221,149,236)(122,225,150,240)(123,214,136,229)(124,218,137,233)(125,222,138,237)(126,211,139,226)(127,215,140,230)(128,219,141,234)(129,223,142,238)(130,212,143,227)(131,216,144,231)(132,220,145,235)(133,224,146,239)(134,213,147,228)(135,217,148,232)(151,204,176,182)(152,208,177,186)(153,197,178,190)(154,201,179,194)(155,205,180,183)(156,209,166,187)(157,198,167,191)(158,202,168,195)(159,206,169,184)(160,210,170,188)(161,199,171,192)(162,203,172,181)(163,207,173,185)(164,196,174,189)(165,200,175,193), (1,166,50,149,25,156,43,121)(2,173,54,147,26,163,32,134)(3,180,58,145,27,155,36,132)(4,172,47,143,28,162,40,130)(5,179,51,141,29,154,44,128)(6,171,55,139,30,161,33,126)(7,178,59,137,16,153,37,124)(8,170,48,150,17,160,41,122)(9,177,52,148,18,152,45,135)(10,169,56,146,19,159,34,133)(11,176,60,144,20,151,38,131)(12,168,49,142,21,158,42,129)(13,175,53,140,22,165,31,127)(14,167,57,138,23,157,35,125)(15,174,46,136,24,164,39,123)(61,234,108,201,80,219,100,194)(62,226,112,199,81,211,104,192)(63,233,116,197,82,218,93,190)(64,240,120,210,83,225,97,188)(65,232,109,208,84,217,101,186)(66,239,113,206,85,224,105,184)(67,231,117,204,86,216,94,182)(68,238,106,202,87,223,98,195)(69,230,110,200,88,215,102,193)(70,237,114,198,89,222,91,191)(71,229,118,196,90,214,95,189)(72,236,107,209,76,221,99,187)(73,228,111,207,77,213,103,185)(74,235,115,205,78,220,92,183)(75,227,119,203,79,212,96,181) );
G=PermutationGroup([[(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)], [(1,76,25,72),(2,87,26,68),(3,83,27,64),(4,79,28,75),(5,90,29,71),(6,86,30,67),(7,82,16,63),(8,78,17,74),(9,89,18,70),(10,85,19,66),(11,81,20,62),(12,77,21,73),(13,88,22,69),(14,84,23,65),(15,80,24,61),(31,110,53,102),(32,106,54,98),(33,117,55,94),(34,113,56,105),(35,109,57,101),(36,120,58,97),(37,116,59,93),(38,112,60,104),(39,108,46,100),(40,119,47,96),(41,115,48,92),(42,111,49,103),(43,107,50,99),(44,118,51,95),(45,114,52,91),(121,187,149,209),(122,183,150,205),(123,194,136,201),(124,190,137,197),(125,186,138,208),(126,182,139,204),(127,193,140,200),(128,189,141,196),(129,185,142,207),(130,181,143,203),(131,192,144,199),(132,188,145,210),(133,184,146,206),(134,195,147,202),(135,191,148,198),(151,211,176,226),(152,222,177,237),(153,218,178,233),(154,214,179,229),(155,225,180,240),(156,221,166,236),(157,217,167,232),(158,213,168,228),(159,224,169,239),(160,220,170,235),(161,216,171,231),(162,212,172,227),(163,223,173,238),(164,219,174,234),(165,215,175,230)], [(1,107,25,99),(2,111,26,103),(3,115,27,92),(4,119,28,96),(5,108,29,100),(6,112,30,104),(7,116,16,93),(8,120,17,97),(9,109,18,101),(10,113,19,105),(11,117,20,94),(12,106,21,98),(13,110,22,102),(14,114,23,91),(15,118,24,95),(31,69,53,88),(32,73,54,77),(33,62,55,81),(34,66,56,85),(35,70,57,89),(36,74,58,78),(37,63,59,82),(38,67,60,86),(39,71,46,90),(40,75,47,79),(41,64,48,83),(42,68,49,87),(43,72,50,76),(44,61,51,80),(45,65,52,84),(121,221,149,236),(122,225,150,240),(123,214,136,229),(124,218,137,233),(125,222,138,237),(126,211,139,226),(127,215,140,230),(128,219,141,234),(129,223,142,238),(130,212,143,227),(131,216,144,231),(132,220,145,235),(133,224,146,239),(134,213,147,228),(135,217,148,232),(151,204,176,182),(152,208,177,186),(153,197,178,190),(154,201,179,194),(155,205,180,183),(156,209,166,187),(157,198,167,191),(158,202,168,195),(159,206,169,184),(160,210,170,188),(161,199,171,192),(162,203,172,181),(163,207,173,185),(164,196,174,189),(165,200,175,193)], [(1,166,50,149,25,156,43,121),(2,173,54,147,26,163,32,134),(3,180,58,145,27,155,36,132),(4,172,47,143,28,162,40,130),(5,179,51,141,29,154,44,128),(6,171,55,139,30,161,33,126),(7,178,59,137,16,153,37,124),(8,170,48,150,17,160,41,122),(9,177,52,148,18,152,45,135),(10,169,56,146,19,159,34,133),(11,176,60,144,20,151,38,131),(12,168,49,142,21,158,42,129),(13,175,53,140,22,165,31,127),(14,167,57,138,23,157,35,125),(15,174,46,136,24,164,39,123),(61,234,108,201,80,219,100,194),(62,226,112,199,81,211,104,192),(63,233,116,197,82,218,93,190),(64,240,120,210,83,225,97,188),(65,232,109,208,84,217,101,186),(66,239,113,206,85,224,105,184),(67,231,117,204,86,216,94,182),(68,238,106,202,87,223,98,195),(69,230,110,200,88,215,102,193),(70,237,114,198,89,222,91,191),(71,229,118,196,90,214,95,189),(72,236,107,209,76,221,99,187),(73,228,111,207,77,213,103,185),(74,235,115,205,78,220,92,183),(75,227,119,203,79,212,96,181)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5 | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 12A | 12B | 12C | 15 | 20 | 24A | 24B | 24C | 24D | 30A | 30B | 30C |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 30 |
| size | 1 | 1 | 2 | 6 | 30 | 2 | 5 | 5 | 6 | 10 | 30 | 4 | 2 | 4 | 10 | 10 | 10 | 10 | 15 | 15 | 15 | 15 | 30 | 30 | 4 | 8 | 24 | 10 | 10 | 20 | 8 | 24 | 20 | 20 | 20 | 20 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | F5 | C2×F5 | C2×F5 | C2×F5 | D12.C4 | S3×F5 | D4.F5 | C2×S3×F5 | D15⋊C8⋊C2 |
| kernel | D15⋊C8⋊C2 | S3×C5⋊C8 | D15⋊C8 | D6.F5 | Dic3.F5 | C3×C22.F5 | C2×C15⋊C8 | Dic3.D10 | C5⋊D12 | C15⋊Q8 | C5×C3⋊D4 | C15⋊7D4 | C22.F5 | C5⋊C8 | C2×Dic5 | Dic5 | C2×C10 | C15 | C3⋊D4 | Dic3 | D6 | C2×C6 | C5 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 |
Matrix representation of D15⋊C8⋊C2 ►in GL8(𝔽241)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 240 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 240 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 0 | 0 | 0 | 0 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 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 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 | 0 | 0 | 0 |
| 64 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 124 | 177 | 198 | 60 |
| 0 | 0 | 0 | 0 | 60 | 134 | 17 | 224 |
| 0 | 0 | 0 | 0 | 17 | 194 | 181 | 107 |
| 0 | 0 | 0 | 0 | 77 | 117 | 64 | 43 |
| 226 | 0 | 182 | 123 | 0 | 0 | 0 | 0 |
| 0 | 226 | 118 | 59 | 0 | 0 | 0 | 0 |
| 59 | 118 | 15 | 0 | 0 | 0 | 0 | 0 |
| 123 | 182 | 0 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 208 | 5 | 13 | 36 |
| 0 | 0 | 0 | 0 | 221 | 41 | 233 | 3 |
| 0 | 0 | 0 | 0 | 200 | 8 | 238 | 16 |
| 0 | 0 | 0 | 0 | 205 | 21 | 33 | 236 |
G:=sub<GL(8,GF(241))| [0,240,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240,240,240,0,0,0,0,1,0,0,0],[0,0,240,1,0,0,0,0,0,0,0,1,0,0,0,0,1,240,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,0,0,124,60,17,77,0,0,0,0,177,134,194,117,0,0,0,0,198,17,181,64,0,0,0,0,60,224,107,43],[226,0,59,123,0,0,0,0,0,226,118,182,0,0,0,0,182,118,15,0,0,0,0,0,123,59,0,15,0,0,0,0,0,0,0,0,208,221,200,205,0,0,0,0,5,41,8,21,0,0,0,0,13,233,238,33,0,0,0,0,36,3,16,236] >;
D15⋊C8⋊C2 in GAP, Magma, Sage, TeX
D_{15}\rtimes C_8\rtimes C_2 % in TeX
G:=Group("D15:C8:C2"); // GroupNames label
G:=SmallGroup(480,1005);
// by ID
G=gap.SmallGroup(480,1005);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,422,219,80,1356,9414,2379]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^4=1,c^2=d^4=b^2,b*a*b^-1=a^11,c*a*c^-1=a^4,d*a*d^-1=a^13,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^2*c>;
// generators/relations