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
Generators and relations for C15⋊Q8.C4
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 [×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
►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 89 23 74)(2 85 24 70)(3 81 25 66)(4 77 26 62)(5 88 27 73)(6 84 28 69)(7 80 29 65)(8 76 30 61)(9 87 16 72)(10 83 17 68)(11 79 18 64)(12 90 19 75)(13 86 20 71)(14 82 21 67)(15 78 22 63)(31 113 50 94)(32 109 51 105)(33 120 52 101)(34 116 53 97)(35 112 54 93)(36 108 55 104)(37 119 56 100)(38 115 57 96)(39 111 58 92)(40 107 59 103)(41 118 60 99)(42 114 46 95)(43 110 47 91)(44 106 48 102)(45 117 49 98)(121 181 142 196)(122 192 143 207)(123 188 144 203)(124 184 145 199)(125 195 146 210)(126 191 147 206)(127 187 148 202)(128 183 149 198)(129 194 150 209)(130 190 136 205)(131 186 137 201)(132 182 138 197)(133 193 139 208)(134 189 140 204)(135 185 141 200)(151 211 166 231)(152 222 167 227)(153 218 168 238)(154 214 169 234)(155 225 170 230)(156 221 171 226)(157 217 172 237)(158 213 173 233)(159 224 174 229)(160 220 175 240)(161 216 176 236)(162 212 177 232)(163 223 178 228)(164 219 179 239)(165 215 180 235)
(1 110 23 91)(2 114 24 95)(3 118 25 99)(4 107 26 103)(5 111 27 92)(6 115 28 96)(7 119 29 100)(8 108 30 104)(9 112 16 93)(10 116 17 97)(11 120 18 101)(12 109 19 105)(13 113 20 94)(14 117 21 98)(15 106 22 102)(31 71 50 86)(32 75 51 90)(33 64 52 79)(34 68 53 83)(35 72 54 87)(36 61 55 76)(37 65 56 80)(38 69 57 84)(39 73 58 88)(40 62 59 77)(41 66 60 81)(42 70 46 85)(43 74 47 89)(44 63 48 78)(45 67 49 82)(121 211 142 231)(122 215 143 235)(123 219 144 239)(124 223 145 228)(125 212 146 232)(126 216 147 236)(127 220 148 240)(128 224 149 229)(129 213 150 233)(130 217 136 237)(131 221 137 226)(132 225 138 230)(133 214 139 234)(134 218 140 238)(135 222 141 227)(151 196 166 181)(152 200 167 185)(153 204 168 189)(154 208 169 193)(155 197 170 182)(156 201 171 186)(157 205 172 190)(158 209 173 194)(159 198 174 183)(160 202 175 187)(161 206 176 191)(162 210 177 195)(163 199 178 184)(164 203 179 188)(165 207 180 192)
(1 166 47 142 23 151 43 121)(2 173 51 140 24 158 32 134)(3 180 55 138 25 165 36 132)(4 172 59 136 26 157 40 130)(5 179 48 149 27 164 44 128)(6 171 52 147 28 156 33 126)(7 178 56 145 29 163 37 124)(8 170 60 143 30 155 41 122)(9 177 49 141 16 162 45 135)(10 169 53 139 17 154 34 133)(11 176 57 137 18 161 38 131)(12 168 46 150 19 153 42 129)(13 175 50 148 20 160 31 127)(14 167 54 146 21 152 35 125)(15 174 58 144 22 159 39 123)(61 230 118 207 76 225 99 192)(62 237 107 205 77 217 103 190)(63 229 111 203 78 224 92 188)(64 236 115 201 79 216 96 186)(65 228 119 199 80 223 100 184)(66 235 108 197 81 215 104 182)(67 227 112 210 82 222 93 195)(68 234 116 208 83 214 97 193)(69 226 120 206 84 221 101 191)(70 233 109 204 85 213 105 189)(71 240 113 202 86 220 94 187)(72 232 117 200 87 212 98 185)(73 239 106 198 88 219 102 183)(74 231 110 196 89 211 91 181)(75 238 114 209 90 218 95 194)
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), (1,89,23,74)(2,85,24,70)(3,81,25,66)(4,77,26,62)(5,88,27,73)(6,84,28,69)(7,80,29,65)(8,76,30,61)(9,87,16,72)(10,83,17,68)(11,79,18,64)(12,90,19,75)(13,86,20,71)(14,82,21,67)(15,78,22,63)(31,113,50,94)(32,109,51,105)(33,120,52,101)(34,116,53,97)(35,112,54,93)(36,108,55,104)(37,119,56,100)(38,115,57,96)(39,111,58,92)(40,107,59,103)(41,118,60,99)(42,114,46,95)(43,110,47,91)(44,106,48,102)(45,117,49,98)(121,181,142,196)(122,192,143,207)(123,188,144,203)(124,184,145,199)(125,195,146,210)(126,191,147,206)(127,187,148,202)(128,183,149,198)(129,194,150,209)(130,190,136,205)(131,186,137,201)(132,182,138,197)(133,193,139,208)(134,189,140,204)(135,185,141,200)(151,211,166,231)(152,222,167,227)(153,218,168,238)(154,214,169,234)(155,225,170,230)(156,221,171,226)(157,217,172,237)(158,213,173,233)(159,224,174,229)(160,220,175,240)(161,216,176,236)(162,212,177,232)(163,223,178,228)(164,219,179,239)(165,215,180,235), (1,110,23,91)(2,114,24,95)(3,118,25,99)(4,107,26,103)(5,111,27,92)(6,115,28,96)(7,119,29,100)(8,108,30,104)(9,112,16,93)(10,116,17,97)(11,120,18,101)(12,109,19,105)(13,113,20,94)(14,117,21,98)(15,106,22,102)(31,71,50,86)(32,75,51,90)(33,64,52,79)(34,68,53,83)(35,72,54,87)(36,61,55,76)(37,65,56,80)(38,69,57,84)(39,73,58,88)(40,62,59,77)(41,66,60,81)(42,70,46,85)(43,74,47,89)(44,63,48,78)(45,67,49,82)(121,211,142,231)(122,215,143,235)(123,219,144,239)(124,223,145,228)(125,212,146,232)(126,216,147,236)(127,220,148,240)(128,224,149,229)(129,213,150,233)(130,217,136,237)(131,221,137,226)(132,225,138,230)(133,214,139,234)(134,218,140,238)(135,222,141,227)(151,196,166,181)(152,200,167,185)(153,204,168,189)(154,208,169,193)(155,197,170,182)(156,201,171,186)(157,205,172,190)(158,209,173,194)(159,198,174,183)(160,202,175,187)(161,206,176,191)(162,210,177,195)(163,199,178,184)(164,203,179,188)(165,207,180,192), (1,166,47,142,23,151,43,121)(2,173,51,140,24,158,32,134)(3,180,55,138,25,165,36,132)(4,172,59,136,26,157,40,130)(5,179,48,149,27,164,44,128)(6,171,52,147,28,156,33,126)(7,178,56,145,29,163,37,124)(8,170,60,143,30,155,41,122)(9,177,49,141,16,162,45,135)(10,169,53,139,17,154,34,133)(11,176,57,137,18,161,38,131)(12,168,46,150,19,153,42,129)(13,175,50,148,20,160,31,127)(14,167,54,146,21,152,35,125)(15,174,58,144,22,159,39,123)(61,230,118,207,76,225,99,192)(62,237,107,205,77,217,103,190)(63,229,111,203,78,224,92,188)(64,236,115,201,79,216,96,186)(65,228,119,199,80,223,100,184)(66,235,108,197,81,215,104,182)(67,227,112,210,82,222,93,195)(68,234,116,208,83,214,97,193)(69,226,120,206,84,221,101,191)(70,233,109,204,85,213,105,189)(71,240,113,202,86,220,94,187)(72,232,117,200,87,212,98,185)(73,239,106,198,88,219,102,183)(74,231,110,196,89,211,91,181)(75,238,114,209,90,218,95,194)>;
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), (1,89,23,74)(2,85,24,70)(3,81,25,66)(4,77,26,62)(5,88,27,73)(6,84,28,69)(7,80,29,65)(8,76,30,61)(9,87,16,72)(10,83,17,68)(11,79,18,64)(12,90,19,75)(13,86,20,71)(14,82,21,67)(15,78,22,63)(31,113,50,94)(32,109,51,105)(33,120,52,101)(34,116,53,97)(35,112,54,93)(36,108,55,104)(37,119,56,100)(38,115,57,96)(39,111,58,92)(40,107,59,103)(41,118,60,99)(42,114,46,95)(43,110,47,91)(44,106,48,102)(45,117,49,98)(121,181,142,196)(122,192,143,207)(123,188,144,203)(124,184,145,199)(125,195,146,210)(126,191,147,206)(127,187,148,202)(128,183,149,198)(129,194,150,209)(130,190,136,205)(131,186,137,201)(132,182,138,197)(133,193,139,208)(134,189,140,204)(135,185,141,200)(151,211,166,231)(152,222,167,227)(153,218,168,238)(154,214,169,234)(155,225,170,230)(156,221,171,226)(157,217,172,237)(158,213,173,233)(159,224,174,229)(160,220,175,240)(161,216,176,236)(162,212,177,232)(163,223,178,228)(164,219,179,239)(165,215,180,235), (1,110,23,91)(2,114,24,95)(3,118,25,99)(4,107,26,103)(5,111,27,92)(6,115,28,96)(7,119,29,100)(8,108,30,104)(9,112,16,93)(10,116,17,97)(11,120,18,101)(12,109,19,105)(13,113,20,94)(14,117,21,98)(15,106,22,102)(31,71,50,86)(32,75,51,90)(33,64,52,79)(34,68,53,83)(35,72,54,87)(36,61,55,76)(37,65,56,80)(38,69,57,84)(39,73,58,88)(40,62,59,77)(41,66,60,81)(42,70,46,85)(43,74,47,89)(44,63,48,78)(45,67,49,82)(121,211,142,231)(122,215,143,235)(123,219,144,239)(124,223,145,228)(125,212,146,232)(126,216,147,236)(127,220,148,240)(128,224,149,229)(129,213,150,233)(130,217,136,237)(131,221,137,226)(132,225,138,230)(133,214,139,234)(134,218,140,238)(135,222,141,227)(151,196,166,181)(152,200,167,185)(153,204,168,189)(154,208,169,193)(155,197,170,182)(156,201,171,186)(157,205,172,190)(158,209,173,194)(159,198,174,183)(160,202,175,187)(161,206,176,191)(162,210,177,195)(163,199,178,184)(164,203,179,188)(165,207,180,192), (1,166,47,142,23,151,43,121)(2,173,51,140,24,158,32,134)(3,180,55,138,25,165,36,132)(4,172,59,136,26,157,40,130)(5,179,48,149,27,164,44,128)(6,171,52,147,28,156,33,126)(7,178,56,145,29,163,37,124)(8,170,60,143,30,155,41,122)(9,177,49,141,16,162,45,135)(10,169,53,139,17,154,34,133)(11,176,57,137,18,161,38,131)(12,168,46,150,19,153,42,129)(13,175,50,148,20,160,31,127)(14,167,54,146,21,152,35,125)(15,174,58,144,22,159,39,123)(61,230,118,207,76,225,99,192)(62,237,107,205,77,217,103,190)(63,229,111,203,78,224,92,188)(64,236,115,201,79,216,96,186)(65,228,119,199,80,223,100,184)(66,235,108,197,81,215,104,182)(67,227,112,210,82,222,93,195)(68,234,116,208,83,214,97,193)(69,226,120,206,84,221,101,191)(70,233,109,204,85,213,105,189)(71,240,113,202,86,220,94,187)(72,232,117,200,87,212,98,185)(73,239,106,198,88,219,102,183)(74,231,110,196,89,211,91,181)(75,238,114,209,90,218,95,194) );
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)], [(1,89,23,74),(2,85,24,70),(3,81,25,66),(4,77,26,62),(5,88,27,73),(6,84,28,69),(7,80,29,65),(8,76,30,61),(9,87,16,72),(10,83,17,68),(11,79,18,64),(12,90,19,75),(13,86,20,71),(14,82,21,67),(15,78,22,63),(31,113,50,94),(32,109,51,105),(33,120,52,101),(34,116,53,97),(35,112,54,93),(36,108,55,104),(37,119,56,100),(38,115,57,96),(39,111,58,92),(40,107,59,103),(41,118,60,99),(42,114,46,95),(43,110,47,91),(44,106,48,102),(45,117,49,98),(121,181,142,196),(122,192,143,207),(123,188,144,203),(124,184,145,199),(125,195,146,210),(126,191,147,206),(127,187,148,202),(128,183,149,198),(129,194,150,209),(130,190,136,205),(131,186,137,201),(132,182,138,197),(133,193,139,208),(134,189,140,204),(135,185,141,200),(151,211,166,231),(152,222,167,227),(153,218,168,238),(154,214,169,234),(155,225,170,230),(156,221,171,226),(157,217,172,237),(158,213,173,233),(159,224,174,229),(160,220,175,240),(161,216,176,236),(162,212,177,232),(163,223,178,228),(164,219,179,239),(165,215,180,235)], [(1,110,23,91),(2,114,24,95),(3,118,25,99),(4,107,26,103),(5,111,27,92),(6,115,28,96),(7,119,29,100),(8,108,30,104),(9,112,16,93),(10,116,17,97),(11,120,18,101),(12,109,19,105),(13,113,20,94),(14,117,21,98),(15,106,22,102),(31,71,50,86),(32,75,51,90),(33,64,52,79),(34,68,53,83),(35,72,54,87),(36,61,55,76),(37,65,56,80),(38,69,57,84),(39,73,58,88),(40,62,59,77),(41,66,60,81),(42,70,46,85),(43,74,47,89),(44,63,48,78),(45,67,49,82),(121,211,142,231),(122,215,143,235),(123,219,144,239),(124,223,145,228),(125,212,146,232),(126,216,147,236),(127,220,148,240),(128,224,149,229),(129,213,150,233),(130,217,136,237),(131,221,137,226),(132,225,138,230),(133,214,139,234),(134,218,140,238),(135,222,141,227),(151,196,166,181),(152,200,167,185),(153,204,168,189),(154,208,169,193),(155,197,170,182),(156,201,171,186),(157,205,172,190),(158,209,173,194),(159,198,174,183),(160,202,175,187),(161,206,176,191),(162,210,177,195),(163,199,178,184),(164,203,179,188),(165,207,180,192)], [(1,166,47,142,23,151,43,121),(2,173,51,140,24,158,32,134),(3,180,55,138,25,165,36,132),(4,172,59,136,26,157,40,130),(5,179,48,149,27,164,44,128),(6,171,52,147,28,156,33,126),(7,178,56,145,29,163,37,124),(8,170,60,143,30,155,41,122),(9,177,49,141,16,162,45,135),(10,169,53,139,17,154,34,133),(11,176,57,137,18,161,38,131),(12,168,46,150,19,153,42,129),(13,175,50,148,20,160,31,127),(14,167,54,146,21,152,35,125),(15,174,58,144,22,159,39,123),(61,230,118,207,76,225,99,192),(62,237,107,205,77,217,103,190),(63,229,111,203,78,224,92,188),(64,236,115,201,79,216,96,186),(65,228,119,199,80,223,100,184),(66,235,108,197,81,215,104,182),(67,227,112,210,82,222,93,195),(68,234,116,208,83,214,97,193),(69,226,120,206,84,221,101,191),(70,233,109,204,85,213,105,189),(71,240,113,202,86,220,94,187),(72,232,117,200,87,212,98,185),(73,239,106,198,88,219,102,183),(74,231,110,196,89,211,91,181),(75,238,114,209,90,218,95,194)])
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 |
Matrix representation of C15⋊Q8.C4 ►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] >;
C15⋊Q8.C4 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