direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C15×C42⋊C2, C42⋊4C30, C4⋊C4⋊6C30, (C2×C4)⋊4C60, (C4×C60)⋊4C2, (C4×C20)⋊8C6, (C2×C12)⋊9C20, (C4×C12)⋊2C10, (C2×C60)⋊29C4, C4.9(C2×C60), (C2×C20)⋊14C12, C60.260(C2×C4), C12.46(C2×C20), C20.67(C2×C12), C22⋊C4.3C30, C2.3(C22×C60), C22.5(C2×C60), C23.9(C2×C30), (C22×C4).6C30, C6.31(C22×C20), (C22×C60).34C2, (C22×C20).18C6, C30.274(C4○D4), (C22×C12).14C10, C10.44(C22×C12), (C2×C60).468C22, C30.238(C22×C4), (C2×C30).452C23, C22.6(C22×C30), (C22×C30).130C22, (C5×C4⋊C4)⋊15C6, (C15×C4⋊C4)⋊33C2, (C3×C4⋊C4)⋊15C10, C2.1(C15×C4○D4), C6.38(C5×C4○D4), (C2×C4).22(C2×C30), (C2×C6).22(C2×C20), (C2×C20).81(C2×C6), C10.38(C3×C4○D4), (C5×C22⋊C4).6C6, (C2×C10).42(C2×C12), (C2×C30).167(C2×C4), (C2×C12).78(C2×C10), (C3×C22⋊C4).6C10, (C15×C22⋊C4).12C2, (C22×C10).33(C2×C6), (C2×C6).72(C22×C10), (C2×C10).72(C22×C6), (C22×C6).25(C2×C10), SmallGroup(480,922)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C42⋊C2
G = < a,b,c,d | a15=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >
Subgroups: 184 in 152 conjugacy classes, 120 normal (32 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, C23, C10, C10, C10, C12, C12, C2×C6, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C2×C12, C2×C12, C22×C6, C30, C30, C30, C42⋊C2, C2×C20, C2×C20, C22×C10, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C60, C60, C2×C30, C2×C30, C2×C30, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C3×C42⋊C2, C2×C60, C2×C60, C22×C30, C5×C42⋊C2, C4×C60, C15×C22⋊C4, C15×C4⋊C4, C22×C60, C15×C42⋊C2
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, C23, C10, C12, C2×C6, C15, C22×C4, C4○D4, C20, C2×C10, C2×C12, C22×C6, C30, C42⋊C2, C2×C20, C22×C10, C22×C12, C3×C4○D4, C60, C2×C30, C22×C20, C5×C4○D4, C3×C42⋊C2, C2×C60, C22×C30, C5×C42⋊C2, C22×C60, C15×C4○D4, C15×C42⋊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 80 32 167)(2 81 33 168)(3 82 34 169)(4 83 35 170)(5 84 36 171)(6 85 37 172)(7 86 38 173)(8 87 39 174)(9 88 40 175)(10 89 41 176)(11 90 42 177)(12 76 43 178)(13 77 44 179)(14 78 45 180)(15 79 31 166)(16 116 220 47)(17 117 221 48)(18 118 222 49)(19 119 223 50)(20 120 224 51)(21 106 225 52)(22 107 211 53)(23 108 212 54)(24 109 213 55)(25 110 214 56)(26 111 215 57)(27 112 216 58)(28 113 217 59)(29 114 218 60)(30 115 219 46)(61 131 93 235)(62 132 94 236)(63 133 95 237)(64 134 96 238)(65 135 97 239)(66 121 98 240)(67 122 99 226)(68 123 100 227)(69 124 101 228)(70 125 102 229)(71 126 103 230)(72 127 104 231)(73 128 105 232)(74 129 91 233)(75 130 92 234)(136 200 152 184)(137 201 153 185)(138 202 154 186)(139 203 155 187)(140 204 156 188)(141 205 157 189)(142 206 158 190)(143 207 159 191)(144 208 160 192)(145 209 161 193)(146 210 162 194)(147 196 163 195)(148 197 164 181)(149 198 165 182)(150 199 151 183)
(1 210 133 213)(2 196 134 214)(3 197 135 215)(4 198 121 216)(5 199 122 217)(6 200 123 218)(7 201 124 219)(8 202 125 220)(9 203 126 221)(10 204 127 222)(11 205 128 223)(12 206 129 224)(13 207 130 225)(14 208 131 211)(15 209 132 212)(16 39 186 229)(17 40 187 230)(18 41 188 231)(19 42 189 232)(20 43 190 233)(21 44 191 234)(22 45 192 235)(23 31 193 236)(24 32 194 237)(25 33 195 238)(26 34 181 239)(27 35 182 240)(28 36 183 226)(29 37 184 227)(30 38 185 228)(46 86 153 101)(47 87 154 102)(48 88 155 103)(49 89 156 104)(50 90 157 105)(51 76 158 91)(52 77 159 92)(53 78 160 93)(54 79 161 94)(55 80 162 95)(56 81 163 96)(57 82 164 97)(58 83 165 98)(59 84 151 99)(60 85 152 100)(61 107 180 144)(62 108 166 145)(63 109 167 146)(64 110 168 147)(65 111 169 148)(66 112 170 149)(67 113 171 150)(68 114 172 136)(69 115 173 137)(70 116 174 138)(71 117 175 139)(72 118 176 140)(73 119 177 141)(74 120 178 142)(75 106 179 143)
(1 32)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 31)(16 220)(17 221)(18 222)(19 223)(20 224)(21 225)(22 211)(23 212)(24 213)(25 214)(26 215)(27 216)(28 217)(29 218)(30 219)(46 137)(47 138)(48 139)(49 140)(50 141)(51 142)(52 143)(53 144)(54 145)(55 146)(56 147)(57 148)(58 149)(59 150)(60 136)(61 78)(62 79)(63 80)(64 81)(65 82)(66 83)(67 84)(68 85)(69 86)(70 87)(71 88)(72 89)(73 90)(74 76)(75 77)(91 178)(92 179)(93 180)(94 166)(95 167)(96 168)(97 169)(98 170)(99 171)(100 172)(101 173)(102 174)(103 175)(104 176)(105 177)(106 159)(107 160)(108 161)(109 162)(110 163)(111 164)(112 165)(113 151)(114 152)(115 153)(116 154)(117 155)(118 156)(119 157)(120 158)(121 240)(122 226)(123 227)(124 228)(125 229)(126 230)(127 231)(128 232)(129 233)(130 234)(131 235)(132 236)(133 237)(134 238)(135 239)(181 197)(182 198)(183 199)(184 200)(185 201)(186 202)(187 203)(188 204)(189 205)(190 206)(191 207)(192 208)(193 209)(194 210)(195 196)
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), (1,80,32,167)(2,81,33,168)(3,82,34,169)(4,83,35,170)(5,84,36,171)(6,85,37,172)(7,86,38,173)(8,87,39,174)(9,88,40,175)(10,89,41,176)(11,90,42,177)(12,76,43,178)(13,77,44,179)(14,78,45,180)(15,79,31,166)(16,116,220,47)(17,117,221,48)(18,118,222,49)(19,119,223,50)(20,120,224,51)(21,106,225,52)(22,107,211,53)(23,108,212,54)(24,109,213,55)(25,110,214,56)(26,111,215,57)(27,112,216,58)(28,113,217,59)(29,114,218,60)(30,115,219,46)(61,131,93,235)(62,132,94,236)(63,133,95,237)(64,134,96,238)(65,135,97,239)(66,121,98,240)(67,122,99,226)(68,123,100,227)(69,124,101,228)(70,125,102,229)(71,126,103,230)(72,127,104,231)(73,128,105,232)(74,129,91,233)(75,130,92,234)(136,200,152,184)(137,201,153,185)(138,202,154,186)(139,203,155,187)(140,204,156,188)(141,205,157,189)(142,206,158,190)(143,207,159,191)(144,208,160,192)(145,209,161,193)(146,210,162,194)(147,196,163,195)(148,197,164,181)(149,198,165,182)(150,199,151,183), (1,210,133,213)(2,196,134,214)(3,197,135,215)(4,198,121,216)(5,199,122,217)(6,200,123,218)(7,201,124,219)(8,202,125,220)(9,203,126,221)(10,204,127,222)(11,205,128,223)(12,206,129,224)(13,207,130,225)(14,208,131,211)(15,209,132,212)(16,39,186,229)(17,40,187,230)(18,41,188,231)(19,42,189,232)(20,43,190,233)(21,44,191,234)(22,45,192,235)(23,31,193,236)(24,32,194,237)(25,33,195,238)(26,34,181,239)(27,35,182,240)(28,36,183,226)(29,37,184,227)(30,38,185,228)(46,86,153,101)(47,87,154,102)(48,88,155,103)(49,89,156,104)(50,90,157,105)(51,76,158,91)(52,77,159,92)(53,78,160,93)(54,79,161,94)(55,80,162,95)(56,81,163,96)(57,82,164,97)(58,83,165,98)(59,84,151,99)(60,85,152,100)(61,107,180,144)(62,108,166,145)(63,109,167,146)(64,110,168,147)(65,111,169,148)(66,112,170,149)(67,113,171,150)(68,114,172,136)(69,115,173,137)(70,116,174,138)(71,117,175,139)(72,118,176,140)(73,119,177,141)(74,120,178,142)(75,106,179,143), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,31)(16,220)(17,221)(18,222)(19,223)(20,224)(21,225)(22,211)(23,212)(24,213)(25,214)(26,215)(27,216)(28,217)(29,218)(30,219)(46,137)(47,138)(48,139)(49,140)(50,141)(51,142)(52,143)(53,144)(54,145)(55,146)(56,147)(57,148)(58,149)(59,150)(60,136)(61,78)(62,79)(63,80)(64,81)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(71,88)(72,89)(73,90)(74,76)(75,77)(91,178)(92,179)(93,180)(94,166)(95,167)(96,168)(97,169)(98,170)(99,171)(100,172)(101,173)(102,174)(103,175)(104,176)(105,177)(106,159)(107,160)(108,161)(109,162)(110,163)(111,164)(112,165)(113,151)(114,152)(115,153)(116,154)(117,155)(118,156)(119,157)(120,158)(121,240)(122,226)(123,227)(124,228)(125,229)(126,230)(127,231)(128,232)(129,233)(130,234)(131,235)(132,236)(133,237)(134,238)(135,239)(181,197)(182,198)(183,199)(184,200)(185,201)(186,202)(187,203)(188,204)(189,205)(190,206)(191,207)(192,208)(193,209)(194,210)(195,196)>;
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), (1,80,32,167)(2,81,33,168)(3,82,34,169)(4,83,35,170)(5,84,36,171)(6,85,37,172)(7,86,38,173)(8,87,39,174)(9,88,40,175)(10,89,41,176)(11,90,42,177)(12,76,43,178)(13,77,44,179)(14,78,45,180)(15,79,31,166)(16,116,220,47)(17,117,221,48)(18,118,222,49)(19,119,223,50)(20,120,224,51)(21,106,225,52)(22,107,211,53)(23,108,212,54)(24,109,213,55)(25,110,214,56)(26,111,215,57)(27,112,216,58)(28,113,217,59)(29,114,218,60)(30,115,219,46)(61,131,93,235)(62,132,94,236)(63,133,95,237)(64,134,96,238)(65,135,97,239)(66,121,98,240)(67,122,99,226)(68,123,100,227)(69,124,101,228)(70,125,102,229)(71,126,103,230)(72,127,104,231)(73,128,105,232)(74,129,91,233)(75,130,92,234)(136,200,152,184)(137,201,153,185)(138,202,154,186)(139,203,155,187)(140,204,156,188)(141,205,157,189)(142,206,158,190)(143,207,159,191)(144,208,160,192)(145,209,161,193)(146,210,162,194)(147,196,163,195)(148,197,164,181)(149,198,165,182)(150,199,151,183), (1,210,133,213)(2,196,134,214)(3,197,135,215)(4,198,121,216)(5,199,122,217)(6,200,123,218)(7,201,124,219)(8,202,125,220)(9,203,126,221)(10,204,127,222)(11,205,128,223)(12,206,129,224)(13,207,130,225)(14,208,131,211)(15,209,132,212)(16,39,186,229)(17,40,187,230)(18,41,188,231)(19,42,189,232)(20,43,190,233)(21,44,191,234)(22,45,192,235)(23,31,193,236)(24,32,194,237)(25,33,195,238)(26,34,181,239)(27,35,182,240)(28,36,183,226)(29,37,184,227)(30,38,185,228)(46,86,153,101)(47,87,154,102)(48,88,155,103)(49,89,156,104)(50,90,157,105)(51,76,158,91)(52,77,159,92)(53,78,160,93)(54,79,161,94)(55,80,162,95)(56,81,163,96)(57,82,164,97)(58,83,165,98)(59,84,151,99)(60,85,152,100)(61,107,180,144)(62,108,166,145)(63,109,167,146)(64,110,168,147)(65,111,169,148)(66,112,170,149)(67,113,171,150)(68,114,172,136)(69,115,173,137)(70,116,174,138)(71,117,175,139)(72,118,176,140)(73,119,177,141)(74,120,178,142)(75,106,179,143), (1,32)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,31)(16,220)(17,221)(18,222)(19,223)(20,224)(21,225)(22,211)(23,212)(24,213)(25,214)(26,215)(27,216)(28,217)(29,218)(30,219)(46,137)(47,138)(48,139)(49,140)(50,141)(51,142)(52,143)(53,144)(54,145)(55,146)(56,147)(57,148)(58,149)(59,150)(60,136)(61,78)(62,79)(63,80)(64,81)(65,82)(66,83)(67,84)(68,85)(69,86)(70,87)(71,88)(72,89)(73,90)(74,76)(75,77)(91,178)(92,179)(93,180)(94,166)(95,167)(96,168)(97,169)(98,170)(99,171)(100,172)(101,173)(102,174)(103,175)(104,176)(105,177)(106,159)(107,160)(108,161)(109,162)(110,163)(111,164)(112,165)(113,151)(114,152)(115,153)(116,154)(117,155)(118,156)(119,157)(120,158)(121,240)(122,226)(123,227)(124,228)(125,229)(126,230)(127,231)(128,232)(129,233)(130,234)(131,235)(132,236)(133,237)(134,238)(135,239)(181,197)(182,198)(183,199)(184,200)(185,201)(186,202)(187,203)(188,204)(189,205)(190,206)(191,207)(192,208)(193,209)(194,210)(195,196) );
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)], [(1,80,32,167),(2,81,33,168),(3,82,34,169),(4,83,35,170),(5,84,36,171),(6,85,37,172),(7,86,38,173),(8,87,39,174),(9,88,40,175),(10,89,41,176),(11,90,42,177),(12,76,43,178),(13,77,44,179),(14,78,45,180),(15,79,31,166),(16,116,220,47),(17,117,221,48),(18,118,222,49),(19,119,223,50),(20,120,224,51),(21,106,225,52),(22,107,211,53),(23,108,212,54),(24,109,213,55),(25,110,214,56),(26,111,215,57),(27,112,216,58),(28,113,217,59),(29,114,218,60),(30,115,219,46),(61,131,93,235),(62,132,94,236),(63,133,95,237),(64,134,96,238),(65,135,97,239),(66,121,98,240),(67,122,99,226),(68,123,100,227),(69,124,101,228),(70,125,102,229),(71,126,103,230),(72,127,104,231),(73,128,105,232),(74,129,91,233),(75,130,92,234),(136,200,152,184),(137,201,153,185),(138,202,154,186),(139,203,155,187),(140,204,156,188),(141,205,157,189),(142,206,158,190),(143,207,159,191),(144,208,160,192),(145,209,161,193),(146,210,162,194),(147,196,163,195),(148,197,164,181),(149,198,165,182),(150,199,151,183)], [(1,210,133,213),(2,196,134,214),(3,197,135,215),(4,198,121,216),(5,199,122,217),(6,200,123,218),(7,201,124,219),(8,202,125,220),(9,203,126,221),(10,204,127,222),(11,205,128,223),(12,206,129,224),(13,207,130,225),(14,208,131,211),(15,209,132,212),(16,39,186,229),(17,40,187,230),(18,41,188,231),(19,42,189,232),(20,43,190,233),(21,44,191,234),(22,45,192,235),(23,31,193,236),(24,32,194,237),(25,33,195,238),(26,34,181,239),(27,35,182,240),(28,36,183,226),(29,37,184,227),(30,38,185,228),(46,86,153,101),(47,87,154,102),(48,88,155,103),(49,89,156,104),(50,90,157,105),(51,76,158,91),(52,77,159,92),(53,78,160,93),(54,79,161,94),(55,80,162,95),(56,81,163,96),(57,82,164,97),(58,83,165,98),(59,84,151,99),(60,85,152,100),(61,107,180,144),(62,108,166,145),(63,109,167,146),(64,110,168,147),(65,111,169,148),(66,112,170,149),(67,113,171,150),(68,114,172,136),(69,115,173,137),(70,116,174,138),(71,117,175,139),(72,118,176,140),(73,119,177,141),(74,120,178,142),(75,106,179,143)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,44),(14,45),(15,31),(16,220),(17,221),(18,222),(19,223),(20,224),(21,225),(22,211),(23,212),(24,213),(25,214),(26,215),(27,216),(28,217),(29,218),(30,219),(46,137),(47,138),(48,139),(49,140),(50,141),(51,142),(52,143),(53,144),(54,145),(55,146),(56,147),(57,148),(58,149),(59,150),(60,136),(61,78),(62,79),(63,80),(64,81),(65,82),(66,83),(67,84),(68,85),(69,86),(70,87),(71,88),(72,89),(73,90),(74,76),(75,77),(91,178),(92,179),(93,180),(94,166),(95,167),(96,168),(97,169),(98,170),(99,171),(100,172),(101,173),(102,174),(103,175),(104,176),(105,177),(106,159),(107,160),(108,161),(109,162),(110,163),(111,164),(112,165),(113,151),(114,152),(115,153),(116,154),(117,155),(118,156),(119,157),(120,158),(121,240),(122,226),(123,227),(124,228),(125,229),(126,230),(127,231),(128,232),(129,233),(130,234),(131,235),(132,236),(133,237),(134,238),(135,239),(181,197),(182,198),(183,199),(184,200),(185,201),(186,202),(187,203),(188,204),(189,205),(190,206),(191,207),(192,208),(193,209),(194,210),(195,196)]])
300 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 10A | ··· | 10L | 10M | ··· | 10T | 12A | ··· | 12H | 12I | ··· | 12AB | 15A | ··· | 15H | 20A | ··· | 20P | 20Q | ··· | 20BD | 30A | ··· | 30X | 30Y | ··· | 30AN | 60A | ··· | 60AF | 60AG | ··· | 60DH |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
300 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C5 | C6 | C6 | C6 | C6 | C10 | C10 | C10 | C10 | C12 | C15 | C20 | C30 | C30 | C30 | C30 | C60 | C4○D4 | C3×C4○D4 | C5×C4○D4 | C15×C4○D4 |
| kernel | C15×C42⋊C2 | C4×C60 | C15×C22⋊C4 | C15×C4⋊C4 | C22×C60 | C5×C42⋊C2 | C2×C60 | C3×C42⋊C2 | C4×C20 | C5×C22⋊C4 | C5×C4⋊C4 | C22×C20 | C4×C12 | C3×C22⋊C4 | C3×C4⋊C4 | C22×C12 | C2×C20 | C42⋊C2 | C2×C12 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C30 | C10 | C6 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 8 | 4 | 4 | 4 | 4 | 2 | 8 | 8 | 8 | 4 | 16 | 8 | 32 | 16 | 16 | 16 | 8 | 64 | 4 | 8 | 16 | 32 |
Matrix representation of C15×C42⋊C2 ►in GL3(𝔽61) generated by
| 1 | 0 | 0 |
| 0 | 22 | 0 |
| 0 | 0 | 22 |
| 11 | 0 | 0 |
| 0 | 53 | 59 |
| 0 | 2 | 8 |
| 1 | 0 | 0 |
| 0 | 11 | 0 |
| 0 | 0 | 11 |
| 1 | 0 | 0 |
| 0 | 60 | 0 |
| 0 | 8 | 1 |
G:=sub<GL(3,GF(61))| [1,0,0,0,22,0,0,0,22],[11,0,0,0,53,2,0,59,8],[1,0,0,0,11,0,0,0,11],[1,0,0,0,60,8,0,0,1] >;
C15×C42⋊C2 in GAP, Magma, Sage, TeX
C_{15}\times C_4^2\rtimes C_2 % in TeX
G:=Group("C15xC4^2:C2"); // GroupNames label
G:=SmallGroup(480,922);
// by ID
G=gap.SmallGroup(480,922);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1680,1709,646]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b*c^2,c*d=d*c>;
// generators/relations