G = C15⋊28(C4×D4) order 480 = 25·3·5
4th semidirect product of C15 and C4×D4 acting via C4×D4/C23=C22
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C15⋊28(C4×D4), C15⋊7D4⋊5C4, D30⋊17(C2×C4), (C5×Dic3)⋊15D4, C30.236(C2×D4), C10.160(S3×D4), C23.D5⋊15S3, D30⋊4C4⋊30C2, C23.31(S3×D5), Dic15⋊12(C2×C4), C6.85(C4○D20), C5⋊7(Dic3⋊4D4), Dic3⋊8(C5⋊D4), Dic15⋊5C4⋊36C2, (C22×Dic3)⋊6D5, (C22×C6).35D10, (Dic3×Dic5)⋊36C2, C30.150(C4○D4), C30.148(C22×C4), (C2×C30).198C23, C22⋊2(D30.C2), (C2×Dic5).130D6, (C22×C10).109D6, C10.57(D4⋊2S3), (C2×Dic3).188D10, C2.7(Dic5.D6), (C22×C30).60C22, (C6×Dic5).114C22, (C22×D15).64C22, (C10×Dic3).203C22, (C2×Dic15).137C22, C3⋊4(C4×C5⋊D4), (C2×C6)⋊2(C4×D5), C6.55(C2×C4×D5), C10.87(S3×C2×C4), C2.7(S3×C5⋊D4), (C2×C30)⋊19(C2×C4), (C2×C10)⋊17(C4×S3), (Dic3×C2×C10)⋊6C2, C6.63(C2×C5⋊D4), C22.88(C2×S3×D5), (C3×C23.D5)⋊9C2, (C2×D30.C2)⋊15C2, (C2×C15⋊7D4).12C2, C2.19(C2×D30.C2), (C2×C6).210(C22×D5), (C2×C10).210(C22×S3), SmallGroup(480,632)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15⋊28(C4×D4)
G = < a,b,c,d | a15=b4=c4=d2=1, bab-1=a11, cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 908 in 188 conjugacy classes, 64 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C2×C4, D4, C23, C23, D5, C10, C10, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, D15, C30, C30, C4×D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C22⋊C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C5×Dic3, C5×Dic3, C3×Dic5, Dic15, D30, D30, C2×C30, C2×C30, C2×C30, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C4×D5, C2×C5⋊D4, C22×C20, Dic3⋊4D4, D30.C2, C6×Dic5, C10×Dic3, C10×Dic3, C2×Dic15, C15⋊7D4, C22×D15, C22×C30, C4×C5⋊D4, Dic3×Dic5, D30⋊4C4, Dic15⋊5C4, C3×C23.D5, C2×D30.C2, Dic3×C2×C10, C2×C15⋊7D4, C15⋊28(C4×D4)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22×C4, C2×D4, C4○D4, D10, C4×S3, C22×S3, C4×D4, C4×D5, C5⋊D4, C22×D5, S3×C2×C4, S3×D4, D4⋊2S3, S3×D5, C2×C4×D5, C4○D20, C2×C5⋊D4, Dic3⋊4D4, D30.C2, C2×S3×D5, C4×C5⋊D4, Dic5.D6, C2×D30.C2, S3×C5⋊D4, C15⋊28(C4×D4)
►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)
(1 155 32 124)(2 151 33 135)(3 162 34 131)(4 158 35 127)(5 154 36 123)(6 165 37 134)(7 161 38 130)(8 157 39 126)(9 153 40 122)(10 164 41 133)(11 160 42 129)(12 156 43 125)(13 152 44 121)(14 163 45 132)(15 159 31 128)(16 172 53 145)(17 168 54 141)(18 179 55 137)(19 175 56 148)(20 171 57 144)(21 167 58 140)(22 178 59 136)(23 174 60 147)(24 170 46 143)(25 166 47 139)(26 177 48 150)(27 173 49 146)(28 169 50 142)(29 180 51 138)(30 176 52 149)(61 223 91 194)(62 219 92 190)(63 215 93 186)(64 211 94 182)(65 222 95 193)(66 218 96 189)(67 214 97 185)(68 225 98 181)(69 221 99 192)(70 217 100 188)(71 213 101 184)(72 224 102 195)(73 220 103 191)(74 216 104 187)(75 212 105 183)(76 234 113 197)(77 230 114 208)(78 226 115 204)(79 237 116 200)(80 233 117 196)(81 229 118 207)(82 240 119 203)(83 236 120 199)(84 232 106 210)(85 228 107 206)(86 239 108 202)(87 235 109 198)(88 231 110 209)(89 227 111 205)(90 238 112 201)
(1 84 24 69)(2 83 25 68)(3 82 26 67)(4 81 27 66)(5 80 28 65)(6 79 29 64)(7 78 30 63)(8 77 16 62)(9 76 17 61)(10 90 18 75)(11 89 19 74)(12 88 20 73)(13 87 21 72)(14 86 22 71)(15 85 23 70)(31 107 60 100)(32 106 46 99)(33 120 47 98)(34 119 48 97)(35 118 49 96)(36 117 50 95)(37 116 51 94)(38 115 52 93)(39 114 53 92)(40 113 54 91)(41 112 55 105)(42 111 56 104)(43 110 57 103)(44 109 58 102)(45 108 59 101)(121 198 140 195)(122 197 141 194)(123 196 142 193)(124 210 143 192)(125 209 144 191)(126 208 145 190)(127 207 146 189)(128 206 147 188)(129 205 148 187)(130 204 149 186)(131 203 150 185)(132 202 136 184)(133 201 137 183)(134 200 138 182)(135 199 139 181)(151 236 166 225)(152 235 167 224)(153 234 168 223)(154 233 169 222)(155 232 170 221)(156 231 171 220)(157 230 172 219)(158 229 173 218)(159 228 174 217)(160 227 175 216)(161 226 176 215)(162 240 177 214)(163 239 178 213)(164 238 179 212)(165 237 180 211)
(1 99)(2 100)(3 101)(4 102)(5 103)(6 104)(7 105)(8 91)(9 92)(10 93)(11 94)(12 95)(13 96)(14 97)(15 98)(16 113)(17 114)(18 115)(19 116)(20 117)(21 118)(22 119)(23 120)(24 106)(25 107)(26 108)(27 109)(28 110)(29 111)(30 112)(31 68)(32 69)(33 70)(34 71)(35 72)(36 73)(37 74)(38 75)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)(45 67)(46 84)(47 85)(48 86)(49 87)(50 88)(51 89)(52 90)(53 76)(54 77)(55 78)(56 79)(57 80)(58 81)(59 82)(60 83)(121 218)(122 219)(123 220)(124 221)(125 222)(126 223)(127 224)(128 225)(129 211)(130 212)(131 213)(132 214)(133 215)(134 216)(135 217)(136 240)(137 226)(138 227)(139 228)(140 229)(141 230)(142 231)(143 232)(144 233)(145 234)(146 235)(147 236)(148 237)(149 238)(150 239)(151 188)(152 189)(153 190)(154 191)(155 192)(156 193)(157 194)(158 195)(159 181)(160 182)(161 183)(162 184)(163 185)(164 186)(165 187)(166 206)(167 207)(168 208)(169 209)(170 210)(171 196)(172 197)(173 198)(174 199)(175 200)(176 201)(177 202)(178 203)(179 204)(180 205)
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), (1,155,32,124)(2,151,33,135)(3,162,34,131)(4,158,35,127)(5,154,36,123)(6,165,37,134)(7,161,38,130)(8,157,39,126)(9,153,40,122)(10,164,41,133)(11,160,42,129)(12,156,43,125)(13,152,44,121)(14,163,45,132)(15,159,31,128)(16,172,53,145)(17,168,54,141)(18,179,55,137)(19,175,56,148)(20,171,57,144)(21,167,58,140)(22,178,59,136)(23,174,60,147)(24,170,46,143)(25,166,47,139)(26,177,48,150)(27,173,49,146)(28,169,50,142)(29,180,51,138)(30,176,52,149)(61,223,91,194)(62,219,92,190)(63,215,93,186)(64,211,94,182)(65,222,95,193)(66,218,96,189)(67,214,97,185)(68,225,98,181)(69,221,99,192)(70,217,100,188)(71,213,101,184)(72,224,102,195)(73,220,103,191)(74,216,104,187)(75,212,105,183)(76,234,113,197)(77,230,114,208)(78,226,115,204)(79,237,116,200)(80,233,117,196)(81,229,118,207)(82,240,119,203)(83,236,120,199)(84,232,106,210)(85,228,107,206)(86,239,108,202)(87,235,109,198)(88,231,110,209)(89,227,111,205)(90,238,112,201), (1,84,24,69)(2,83,25,68)(3,82,26,67)(4,81,27,66)(5,80,28,65)(6,79,29,64)(7,78,30,63)(8,77,16,62)(9,76,17,61)(10,90,18,75)(11,89,19,74)(12,88,20,73)(13,87,21,72)(14,86,22,71)(15,85,23,70)(31,107,60,100)(32,106,46,99)(33,120,47,98)(34,119,48,97)(35,118,49,96)(36,117,50,95)(37,116,51,94)(38,115,52,93)(39,114,53,92)(40,113,54,91)(41,112,55,105)(42,111,56,104)(43,110,57,103)(44,109,58,102)(45,108,59,101)(121,198,140,195)(122,197,141,194)(123,196,142,193)(124,210,143,192)(125,209,144,191)(126,208,145,190)(127,207,146,189)(128,206,147,188)(129,205,148,187)(130,204,149,186)(131,203,150,185)(132,202,136,184)(133,201,137,183)(134,200,138,182)(135,199,139,181)(151,236,166,225)(152,235,167,224)(153,234,168,223)(154,233,169,222)(155,232,170,221)(156,231,171,220)(157,230,172,219)(158,229,173,218)(159,228,174,217)(160,227,175,216)(161,226,176,215)(162,240,177,214)(163,239,178,213)(164,238,179,212)(165,237,180,211), (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,105)(8,91)(9,92)(10,93)(11,94)(12,95)(13,96)(14,97)(15,98)(16,113)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(38,75)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66)(45,67)(46,84)(47,85)(48,86)(49,87)(50,88)(51,89)(52,90)(53,76)(54,77)(55,78)(56,79)(57,80)(58,81)(59,82)(60,83)(121,218)(122,219)(123,220)(124,221)(125,222)(126,223)(127,224)(128,225)(129,211)(130,212)(131,213)(132,214)(133,215)(134,216)(135,217)(136,240)(137,226)(138,227)(139,228)(140,229)(141,230)(142,231)(143,232)(144,233)(145,234)(146,235)(147,236)(148,237)(149,238)(150,239)(151,188)(152,189)(153,190)(154,191)(155,192)(156,193)(157,194)(158,195)(159,181)(160,182)(161,183)(162,184)(163,185)(164,186)(165,187)(166,206)(167,207)(168,208)(169,209)(170,210)(171,196)(172,197)(173,198)(174,199)(175,200)(176,201)(177,202)(178,203)(179,204)(180,205)>;
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), (1,155,32,124)(2,151,33,135)(3,162,34,131)(4,158,35,127)(5,154,36,123)(6,165,37,134)(7,161,38,130)(8,157,39,126)(9,153,40,122)(10,164,41,133)(11,160,42,129)(12,156,43,125)(13,152,44,121)(14,163,45,132)(15,159,31,128)(16,172,53,145)(17,168,54,141)(18,179,55,137)(19,175,56,148)(20,171,57,144)(21,167,58,140)(22,178,59,136)(23,174,60,147)(24,170,46,143)(25,166,47,139)(26,177,48,150)(27,173,49,146)(28,169,50,142)(29,180,51,138)(30,176,52,149)(61,223,91,194)(62,219,92,190)(63,215,93,186)(64,211,94,182)(65,222,95,193)(66,218,96,189)(67,214,97,185)(68,225,98,181)(69,221,99,192)(70,217,100,188)(71,213,101,184)(72,224,102,195)(73,220,103,191)(74,216,104,187)(75,212,105,183)(76,234,113,197)(77,230,114,208)(78,226,115,204)(79,237,116,200)(80,233,117,196)(81,229,118,207)(82,240,119,203)(83,236,120,199)(84,232,106,210)(85,228,107,206)(86,239,108,202)(87,235,109,198)(88,231,110,209)(89,227,111,205)(90,238,112,201), (1,84,24,69)(2,83,25,68)(3,82,26,67)(4,81,27,66)(5,80,28,65)(6,79,29,64)(7,78,30,63)(8,77,16,62)(9,76,17,61)(10,90,18,75)(11,89,19,74)(12,88,20,73)(13,87,21,72)(14,86,22,71)(15,85,23,70)(31,107,60,100)(32,106,46,99)(33,120,47,98)(34,119,48,97)(35,118,49,96)(36,117,50,95)(37,116,51,94)(38,115,52,93)(39,114,53,92)(40,113,54,91)(41,112,55,105)(42,111,56,104)(43,110,57,103)(44,109,58,102)(45,108,59,101)(121,198,140,195)(122,197,141,194)(123,196,142,193)(124,210,143,192)(125,209,144,191)(126,208,145,190)(127,207,146,189)(128,206,147,188)(129,205,148,187)(130,204,149,186)(131,203,150,185)(132,202,136,184)(133,201,137,183)(134,200,138,182)(135,199,139,181)(151,236,166,225)(152,235,167,224)(153,234,168,223)(154,233,169,222)(155,232,170,221)(156,231,171,220)(157,230,172,219)(158,229,173,218)(159,228,174,217)(160,227,175,216)(161,226,176,215)(162,240,177,214)(163,239,178,213)(164,238,179,212)(165,237,180,211), (1,99)(2,100)(3,101)(4,102)(5,103)(6,104)(7,105)(8,91)(9,92)(10,93)(11,94)(12,95)(13,96)(14,97)(15,98)(16,113)(17,114)(18,115)(19,116)(20,117)(21,118)(22,119)(23,120)(24,106)(25,107)(26,108)(27,109)(28,110)(29,111)(30,112)(31,68)(32,69)(33,70)(34,71)(35,72)(36,73)(37,74)(38,75)(39,61)(40,62)(41,63)(42,64)(43,65)(44,66)(45,67)(46,84)(47,85)(48,86)(49,87)(50,88)(51,89)(52,90)(53,76)(54,77)(55,78)(56,79)(57,80)(58,81)(59,82)(60,83)(121,218)(122,219)(123,220)(124,221)(125,222)(126,223)(127,224)(128,225)(129,211)(130,212)(131,213)(132,214)(133,215)(134,216)(135,217)(136,240)(137,226)(138,227)(139,228)(140,229)(141,230)(142,231)(143,232)(144,233)(145,234)(146,235)(147,236)(148,237)(149,238)(150,239)(151,188)(152,189)(153,190)(154,191)(155,192)(156,193)(157,194)(158,195)(159,181)(160,182)(161,183)(162,184)(163,185)(164,186)(165,187)(166,206)(167,207)(168,208)(169,209)(170,210)(171,196)(172,197)(173,198)(174,199)(175,200)(176,201)(177,202)(178,203)(179,204)(180,205) );
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)], [(1,155,32,124),(2,151,33,135),(3,162,34,131),(4,158,35,127),(5,154,36,123),(6,165,37,134),(7,161,38,130),(8,157,39,126),(9,153,40,122),(10,164,41,133),(11,160,42,129),(12,156,43,125),(13,152,44,121),(14,163,45,132),(15,159,31,128),(16,172,53,145),(17,168,54,141),(18,179,55,137),(19,175,56,148),(20,171,57,144),(21,167,58,140),(22,178,59,136),(23,174,60,147),(24,170,46,143),(25,166,47,139),(26,177,48,150),(27,173,49,146),(28,169,50,142),(29,180,51,138),(30,176,52,149),(61,223,91,194),(62,219,92,190),(63,215,93,186),(64,211,94,182),(65,222,95,193),(66,218,96,189),(67,214,97,185),(68,225,98,181),(69,221,99,192),(70,217,100,188),(71,213,101,184),(72,224,102,195),(73,220,103,191),(74,216,104,187),(75,212,105,183),(76,234,113,197),(77,230,114,208),(78,226,115,204),(79,237,116,200),(80,233,117,196),(81,229,118,207),(82,240,119,203),(83,236,120,199),(84,232,106,210),(85,228,107,206),(86,239,108,202),(87,235,109,198),(88,231,110,209),(89,227,111,205),(90,238,112,201)], [(1,84,24,69),(2,83,25,68),(3,82,26,67),(4,81,27,66),(5,80,28,65),(6,79,29,64),(7,78,30,63),(8,77,16,62),(9,76,17,61),(10,90,18,75),(11,89,19,74),(12,88,20,73),(13,87,21,72),(14,86,22,71),(15,85,23,70),(31,107,60,100),(32,106,46,99),(33,120,47,98),(34,119,48,97),(35,118,49,96),(36,117,50,95),(37,116,51,94),(38,115,52,93),(39,114,53,92),(40,113,54,91),(41,112,55,105),(42,111,56,104),(43,110,57,103),(44,109,58,102),(45,108,59,101),(121,198,140,195),(122,197,141,194),(123,196,142,193),(124,210,143,192),(125,209,144,191),(126,208,145,190),(127,207,146,189),(128,206,147,188),(129,205,148,187),(130,204,149,186),(131,203,150,185),(132,202,136,184),(133,201,137,183),(134,200,138,182),(135,199,139,181),(151,236,166,225),(152,235,167,224),(153,234,168,223),(154,233,169,222),(155,232,170,221),(156,231,171,220),(157,230,172,219),(158,229,173,218),(159,228,174,217),(160,227,175,216),(161,226,176,215),(162,240,177,214),(163,239,178,213),(164,238,179,212),(165,237,180,211)], [(1,99),(2,100),(3,101),(4,102),(5,103),(6,104),(7,105),(8,91),(9,92),(10,93),(11,94),(12,95),(13,96),(14,97),(15,98),(16,113),(17,114),(18,115),(19,116),(20,117),(21,118),(22,119),(23,120),(24,106),(25,107),(26,108),(27,109),(28,110),(29,111),(30,112),(31,68),(32,69),(33,70),(34,71),(35,72),(36,73),(37,74),(38,75),(39,61),(40,62),(41,63),(42,64),(43,65),(44,66),(45,67),(46,84),(47,85),(48,86),(49,87),(50,88),(51,89),(52,90),(53,76),(54,77),(55,78),(56,79),(57,80),(58,81),(59,82),(60,83),(121,218),(122,219),(123,220),(124,221),(125,222),(126,223),(127,224),(128,225),(129,211),(130,212),(131,213),(132,214),(133,215),(134,216),(135,217),(136,240),(137,226),(138,227),(139,228),(140,229),(141,230),(142,231),(143,232),(144,233),(145,234),(146,235),(147,236),(148,237),(149,238),(150,239),(151,188),(152,189),(153,190),(154,191),(155,192),(156,193),(157,194),(158,195),(159,181),(160,182),(161,183),(162,184),(163,185),(164,186),(165,187),(166,206),(167,207),(168,208),(169,209),(170,210),(171,196),(172,197),(173,198),(174,199),(175,200),(176,201),(177,202),(178,203),(179,204),(180,205)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10N | 12A | 12B | 12C | 12D | 15A | 15B | 20A | ··· | 20P | 30A | ··· | 30N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 30 | 30 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 10 | 10 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 20 | 20 | 20 | 20 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C4×S3 | C5⋊D4 | C4×D5 | C4○D20 | S3×D4 | D4⋊2S3 | S3×D5 | D30.C2 | C2×S3×D5 | Dic5.D6 | S3×C5⋊D4 |
| kernel | C15⋊28(C4×D4) | Dic3×Dic5 | D30⋊4C4 | Dic15⋊5C4 | C3×C23.D5 | C2×D30.C2 | Dic3×C2×C10 | C2×C15⋊7D4 | C15⋊7D4 | C23.D5 | C5×Dic3 | C22×Dic3 | C2×Dic5 | C22×C10 | C30 | C2×Dic3 | C22×C6 | C2×C10 | Dic3 | C2×C6 | C6 | C10 | C10 | C23 | C22 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 8 | 8 | 8 | 1 | 1 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C15⋊28(C4×D4) ►in GL6(𝔽61)
| 60 | 17 | 0 | 0 | 0 | 0 |
| 44 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 43 | 0 | 0 |
| 0 | 0 | 17 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 0 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 14 | 45 | 0 | 0 | 0 | 0 |
| 39 | 47 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 18 | 0 | 0 |
| 0 | 0 | 45 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 47 | 16 | 0 | 0 | 0 | 0 |
| 45 | 14 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(61))| [60,44,0,0,0,0,17,44,0,0,0,0,0,0,0,17,0,0,0,0,43,17,0,0,0,0,0,0,60,60,0,0,0,0,1,0],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,11,0,0,0,0,0,0,11,0,0,0,0,0,0,0,60,0,0,0,0,60,0],[14,39,0,0,0,0,45,47,0,0,0,0,0,0,17,45,0,0,0,0,18,44,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[47,45,0,0,0,0,16,14,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C15⋊28(C4×D4) in GAP, Magma, Sage, TeX
C_{15}\rtimes_{28}(C_4\times D_4) % in TeX
G:=Group("C15:28(C4xD4)"); // GroupNames label
G:=SmallGroup(480,632);
// by ID
G=gap.SmallGroup(480,632);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,422,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^4=c^4=d^2=1,b*a*b^-1=a^11,c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations