metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊C8.3D6, C3⋊D4.F5, C15⋊Q8.2C4, D15⋊C8⋊5C2, C15⋊6(C8○D4), D6.F5⋊6C2, C3⋊4(D4.F5), D6.4(C2×F5), C5⋊3(D12.C4), D30.4(C2×C4), C5⋊D12.2C4, C15⋊7D4.2C4, 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, Dic15.6(C2×C4), (C2×Dic5).76D6, 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
Subgroups: 564 in 124 conjugacy classes, 46 normal (all characteristic)
C1, C2, C2 [×3], C3, C4 [×4], C22, C22 [×2], C5, S3 [×2], C6, C6, C8 [×4], C2×C4 [×3], D4 [×3], Q8, D5, C10, C10 [×2], Dic3, Dic3, C12 [×2], D6, D6, C2×C6, C15, C2×C8 [×3], M4(2) [×3], C4○D4, Dic5 [×2], Dic5, C20, D10, C2×C10, C2×C10, C3⋊C8 [×2], C24 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4, C3⋊D4, C2×C12, C5×S3, D15, C30, C30, C8○D4, C5⋊C8 [×2], C5⋊C8 [×2], Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4 [×2], C5×D4, S3×C8 [×2], C8⋊S3 [×2], C2×C3⋊C8, C3×M4(2), C4○D12, C5×Dic3, C3×Dic5 [×2], Dic15, S3×C10, D30, C2×C30, D5⋊C8, C4.F5, C2×C5⋊C8 [×2], C22.F5, C22.F5, D4⋊2D5, D12.C4, C3×C5⋊C8 [×2], C15⋊C8 [×2], 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, C15⋊Q8.C4
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], C22×C4, F5, C4×S3 [×2], C22×S3, C8○D4, C2×F5 [×3], S3×C2×C4, C22×F5, D12.C4, S3×F5, D4.F5, C2×S3×F5, C15⋊Q8.C4
Generators and relations
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 >
►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 85 16 70)(2 81 17 66)(3 77 18 62)(4 88 19 73)(5 84 20 69)(6 80 21 65)(7 76 22 61)(8 87 23 72)(9 83 24 68)(10 79 25 64)(11 90 26 75)(12 86 27 71)(13 82 28 67)(14 78 29 63)(15 89 30 74)(31 110 57 94)(32 106 58 105)(33 117 59 101)(34 113 60 97)(35 109 46 93)(36 120 47 104)(37 116 48 100)(38 112 49 96)(39 108 50 92)(40 119 51 103)(41 115 52 99)(42 111 53 95)(43 107 54 91)(44 118 55 102)(45 114 56 98)(121 195 145 202)(122 191 146 198)(123 187 147 209)(124 183 148 205)(125 194 149 201)(126 190 150 197)(127 186 136 208)(128 182 137 204)(129 193 138 200)(130 189 139 196)(131 185 140 207)(132 181 141 203)(133 192 142 199)(134 188 143 210)(135 184 144 206)(151 216 171 231)(152 212 172 227)(153 223 173 238)(154 219 174 234)(155 215 175 230)(156 211 176 226)(157 222 177 237)(158 218 178 233)(159 214 179 229)(160 225 180 240)(161 221 166 236)(162 217 167 232)(163 213 168 228)(164 224 169 239)(165 220 170 235)
(1 107 16 91)(2 111 17 95)(3 115 18 99)(4 119 19 103)(5 108 20 92)(6 112 21 96)(7 116 22 100)(8 120 23 104)(9 109 24 93)(10 113 25 97)(11 117 26 101)(12 106 27 105)(13 110 28 94)(14 114 29 98)(15 118 30 102)(31 67 57 82)(32 71 58 86)(33 75 59 90)(34 64 60 79)(35 68 46 83)(36 72 47 87)(37 61 48 76)(38 65 49 80)(39 69 50 84)(40 73 51 88)(41 62 52 77)(42 66 53 81)(43 70 54 85)(44 74 55 89)(45 63 56 78)(121 214 145 229)(122 218 146 233)(123 222 147 237)(124 211 148 226)(125 215 149 230)(126 219 150 234)(127 223 136 238)(128 212 137 227)(129 216 138 231)(130 220 139 235)(131 224 140 239)(132 213 141 228)(133 217 142 232)(134 221 143 236)(135 225 144 240)(151 200 171 193)(152 204 172 182)(153 208 173 186)(154 197 174 190)(155 201 175 194)(156 205 176 183)(157 209 177 187)(158 198 178 191)(159 202 179 195)(160 206 180 184)(161 210 166 188)(162 199 167 192)(163 203 168 181)(164 207 169 185)(165 196 170 189)
(1 166 54 143 16 161 43 134)(2 173 58 141 17 153 32 132)(3 180 47 139 18 160 36 130)(4 172 51 137 19 152 40 128)(5 179 55 150 20 159 44 126)(6 171 59 148 21 151 33 124)(7 178 48 146 22 158 37 122)(8 170 52 144 23 165 41 135)(9 177 56 142 24 157 45 133)(10 169 60 140 25 164 34 131)(11 176 49 138 26 156 38 129)(12 168 53 136 27 163 42 127)(13 175 57 149 28 155 31 125)(14 167 46 147 29 162 35 123)(15 174 50 145 30 154 39 121)(61 233 116 198 76 218 100 191)(62 240 120 196 77 225 104 189)(63 232 109 209 78 217 93 187)(64 239 113 207 79 224 97 185)(65 231 117 205 80 216 101 183)(66 238 106 203 81 223 105 181)(67 230 110 201 82 215 94 194)(68 237 114 199 83 222 98 192)(69 229 118 197 84 214 102 190)(70 236 107 210 85 221 91 188)(71 228 111 208 86 213 95 186)(72 235 115 206 87 220 99 184)(73 227 119 204 88 212 103 182)(74 234 108 202 89 219 92 195)(75 226 112 200 90 211 96 193)
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), (1,85,16,70)(2,81,17,66)(3,77,18,62)(4,88,19,73)(5,84,20,69)(6,80,21,65)(7,76,22,61)(8,87,23,72)(9,83,24,68)(10,79,25,64)(11,90,26,75)(12,86,27,71)(13,82,28,67)(14,78,29,63)(15,89,30,74)(31,110,57,94)(32,106,58,105)(33,117,59,101)(34,113,60,97)(35,109,46,93)(36,120,47,104)(37,116,48,100)(38,112,49,96)(39,108,50,92)(40,119,51,103)(41,115,52,99)(42,111,53,95)(43,107,54,91)(44,118,55,102)(45,114,56,98)(121,195,145,202)(122,191,146,198)(123,187,147,209)(124,183,148,205)(125,194,149,201)(126,190,150,197)(127,186,136,208)(128,182,137,204)(129,193,138,200)(130,189,139,196)(131,185,140,207)(132,181,141,203)(133,192,142,199)(134,188,143,210)(135,184,144,206)(151,216,171,231)(152,212,172,227)(153,223,173,238)(154,219,174,234)(155,215,175,230)(156,211,176,226)(157,222,177,237)(158,218,178,233)(159,214,179,229)(160,225,180,240)(161,221,166,236)(162,217,167,232)(163,213,168,228)(164,224,169,239)(165,220,170,235), (1,107,16,91)(2,111,17,95)(3,115,18,99)(4,119,19,103)(5,108,20,92)(6,112,21,96)(7,116,22,100)(8,120,23,104)(9,109,24,93)(10,113,25,97)(11,117,26,101)(12,106,27,105)(13,110,28,94)(14,114,29,98)(15,118,30,102)(31,67,57,82)(32,71,58,86)(33,75,59,90)(34,64,60,79)(35,68,46,83)(36,72,47,87)(37,61,48,76)(38,65,49,80)(39,69,50,84)(40,73,51,88)(41,62,52,77)(42,66,53,81)(43,70,54,85)(44,74,55,89)(45,63,56,78)(121,214,145,229)(122,218,146,233)(123,222,147,237)(124,211,148,226)(125,215,149,230)(126,219,150,234)(127,223,136,238)(128,212,137,227)(129,216,138,231)(130,220,139,235)(131,224,140,239)(132,213,141,228)(133,217,142,232)(134,221,143,236)(135,225,144,240)(151,200,171,193)(152,204,172,182)(153,208,173,186)(154,197,174,190)(155,201,175,194)(156,205,176,183)(157,209,177,187)(158,198,178,191)(159,202,179,195)(160,206,180,184)(161,210,166,188)(162,199,167,192)(163,203,168,181)(164,207,169,185)(165,196,170,189), (1,166,54,143,16,161,43,134)(2,173,58,141,17,153,32,132)(3,180,47,139,18,160,36,130)(4,172,51,137,19,152,40,128)(5,179,55,150,20,159,44,126)(6,171,59,148,21,151,33,124)(7,178,48,146,22,158,37,122)(8,170,52,144,23,165,41,135)(9,177,56,142,24,157,45,133)(10,169,60,140,25,164,34,131)(11,176,49,138,26,156,38,129)(12,168,53,136,27,163,42,127)(13,175,57,149,28,155,31,125)(14,167,46,147,29,162,35,123)(15,174,50,145,30,154,39,121)(61,233,116,198,76,218,100,191)(62,240,120,196,77,225,104,189)(63,232,109,209,78,217,93,187)(64,239,113,207,79,224,97,185)(65,231,117,205,80,216,101,183)(66,238,106,203,81,223,105,181)(67,230,110,201,82,215,94,194)(68,237,114,199,83,222,98,192)(69,229,118,197,84,214,102,190)(70,236,107,210,85,221,91,188)(71,228,111,208,86,213,95,186)(72,235,115,206,87,220,99,184)(73,227,119,204,88,212,103,182)(74,234,108,202,89,219,92,195)(75,226,112,200,90,211,96,193)>;
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), (1,85,16,70)(2,81,17,66)(3,77,18,62)(4,88,19,73)(5,84,20,69)(6,80,21,65)(7,76,22,61)(8,87,23,72)(9,83,24,68)(10,79,25,64)(11,90,26,75)(12,86,27,71)(13,82,28,67)(14,78,29,63)(15,89,30,74)(31,110,57,94)(32,106,58,105)(33,117,59,101)(34,113,60,97)(35,109,46,93)(36,120,47,104)(37,116,48,100)(38,112,49,96)(39,108,50,92)(40,119,51,103)(41,115,52,99)(42,111,53,95)(43,107,54,91)(44,118,55,102)(45,114,56,98)(121,195,145,202)(122,191,146,198)(123,187,147,209)(124,183,148,205)(125,194,149,201)(126,190,150,197)(127,186,136,208)(128,182,137,204)(129,193,138,200)(130,189,139,196)(131,185,140,207)(132,181,141,203)(133,192,142,199)(134,188,143,210)(135,184,144,206)(151,216,171,231)(152,212,172,227)(153,223,173,238)(154,219,174,234)(155,215,175,230)(156,211,176,226)(157,222,177,237)(158,218,178,233)(159,214,179,229)(160,225,180,240)(161,221,166,236)(162,217,167,232)(163,213,168,228)(164,224,169,239)(165,220,170,235), (1,107,16,91)(2,111,17,95)(3,115,18,99)(4,119,19,103)(5,108,20,92)(6,112,21,96)(7,116,22,100)(8,120,23,104)(9,109,24,93)(10,113,25,97)(11,117,26,101)(12,106,27,105)(13,110,28,94)(14,114,29,98)(15,118,30,102)(31,67,57,82)(32,71,58,86)(33,75,59,90)(34,64,60,79)(35,68,46,83)(36,72,47,87)(37,61,48,76)(38,65,49,80)(39,69,50,84)(40,73,51,88)(41,62,52,77)(42,66,53,81)(43,70,54,85)(44,74,55,89)(45,63,56,78)(121,214,145,229)(122,218,146,233)(123,222,147,237)(124,211,148,226)(125,215,149,230)(126,219,150,234)(127,223,136,238)(128,212,137,227)(129,216,138,231)(130,220,139,235)(131,224,140,239)(132,213,141,228)(133,217,142,232)(134,221,143,236)(135,225,144,240)(151,200,171,193)(152,204,172,182)(153,208,173,186)(154,197,174,190)(155,201,175,194)(156,205,176,183)(157,209,177,187)(158,198,178,191)(159,202,179,195)(160,206,180,184)(161,210,166,188)(162,199,167,192)(163,203,168,181)(164,207,169,185)(165,196,170,189), (1,166,54,143,16,161,43,134)(2,173,58,141,17,153,32,132)(3,180,47,139,18,160,36,130)(4,172,51,137,19,152,40,128)(5,179,55,150,20,159,44,126)(6,171,59,148,21,151,33,124)(7,178,48,146,22,158,37,122)(8,170,52,144,23,165,41,135)(9,177,56,142,24,157,45,133)(10,169,60,140,25,164,34,131)(11,176,49,138,26,156,38,129)(12,168,53,136,27,163,42,127)(13,175,57,149,28,155,31,125)(14,167,46,147,29,162,35,123)(15,174,50,145,30,154,39,121)(61,233,116,198,76,218,100,191)(62,240,120,196,77,225,104,189)(63,232,109,209,78,217,93,187)(64,239,113,207,79,224,97,185)(65,231,117,205,80,216,101,183)(66,238,106,203,81,223,105,181)(67,230,110,201,82,215,94,194)(68,237,114,199,83,222,98,192)(69,229,118,197,84,214,102,190)(70,236,107,210,85,221,91,188)(71,228,111,208,86,213,95,186)(72,235,115,206,87,220,99,184)(73,227,119,204,88,212,103,182)(74,234,108,202,89,219,92,195)(75,226,112,200,90,211,96,193) );
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)], [(1,85,16,70),(2,81,17,66),(3,77,18,62),(4,88,19,73),(5,84,20,69),(6,80,21,65),(7,76,22,61),(8,87,23,72),(9,83,24,68),(10,79,25,64),(11,90,26,75),(12,86,27,71),(13,82,28,67),(14,78,29,63),(15,89,30,74),(31,110,57,94),(32,106,58,105),(33,117,59,101),(34,113,60,97),(35,109,46,93),(36,120,47,104),(37,116,48,100),(38,112,49,96),(39,108,50,92),(40,119,51,103),(41,115,52,99),(42,111,53,95),(43,107,54,91),(44,118,55,102),(45,114,56,98),(121,195,145,202),(122,191,146,198),(123,187,147,209),(124,183,148,205),(125,194,149,201),(126,190,150,197),(127,186,136,208),(128,182,137,204),(129,193,138,200),(130,189,139,196),(131,185,140,207),(132,181,141,203),(133,192,142,199),(134,188,143,210),(135,184,144,206),(151,216,171,231),(152,212,172,227),(153,223,173,238),(154,219,174,234),(155,215,175,230),(156,211,176,226),(157,222,177,237),(158,218,178,233),(159,214,179,229),(160,225,180,240),(161,221,166,236),(162,217,167,232),(163,213,168,228),(164,224,169,239),(165,220,170,235)], [(1,107,16,91),(2,111,17,95),(3,115,18,99),(4,119,19,103),(5,108,20,92),(6,112,21,96),(7,116,22,100),(8,120,23,104),(9,109,24,93),(10,113,25,97),(11,117,26,101),(12,106,27,105),(13,110,28,94),(14,114,29,98),(15,118,30,102),(31,67,57,82),(32,71,58,86),(33,75,59,90),(34,64,60,79),(35,68,46,83),(36,72,47,87),(37,61,48,76),(38,65,49,80),(39,69,50,84),(40,73,51,88),(41,62,52,77),(42,66,53,81),(43,70,54,85),(44,74,55,89),(45,63,56,78),(121,214,145,229),(122,218,146,233),(123,222,147,237),(124,211,148,226),(125,215,149,230),(126,219,150,234),(127,223,136,238),(128,212,137,227),(129,216,138,231),(130,220,139,235),(131,224,140,239),(132,213,141,228),(133,217,142,232),(134,221,143,236),(135,225,144,240),(151,200,171,193),(152,204,172,182),(153,208,173,186),(154,197,174,190),(155,201,175,194),(156,205,176,183),(157,209,177,187),(158,198,178,191),(159,202,179,195),(160,206,180,184),(161,210,166,188),(162,199,167,192),(163,203,168,181),(164,207,169,185),(165,196,170,189)], [(1,166,54,143,16,161,43,134),(2,173,58,141,17,153,32,132),(3,180,47,139,18,160,36,130),(4,172,51,137,19,152,40,128),(5,179,55,150,20,159,44,126),(6,171,59,148,21,151,33,124),(7,178,48,146,22,158,37,122),(8,170,52,144,23,165,41,135),(9,177,56,142,24,157,45,133),(10,169,60,140,25,164,34,131),(11,176,49,138,26,156,38,129),(12,168,53,136,27,163,42,127),(13,175,57,149,28,155,31,125),(14,167,46,147,29,162,35,123),(15,174,50,145,30,154,39,121),(61,233,116,198,76,218,100,191),(62,240,120,196,77,225,104,189),(63,232,109,209,78,217,93,187),(64,239,113,207,79,224,97,185),(65,231,117,205,80,216,101,183),(66,238,106,203,81,223,105,181),(67,230,110,201,82,215,94,194),(68,237,114,199,83,222,98,192),(69,229,118,197,84,214,102,190),(70,236,107,210,85,221,91,188),(71,228,111,208,86,213,95,186),(72,235,115,206,87,220,99,184),(73,227,119,204,88,212,103,182),(74,234,108,202,89,219,92,195),(75,226,112,200,90,211,96,193)])
Matrix representation ►G ⊆ 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] >;
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 | C15⋊Q8.C4 |
| kernel | C15⋊Q8.C4 | 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 |
In GAP, Magma, Sage, TeX
C_{15}\rtimes Q_8.C_4 % in TeX
G:=Group("C15:Q8.C4"); // 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
×
⋊
ℤ
𝔽
○
≀