direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C15×C4.4D4, C42⋊8C30, C60.196D4, (C4×C20)⋊18C6, (C4×C60)⋊26C2, (C6×Q8)⋊9C10, (C2×Q8)⋊4C30, C4.4(D4×C15), C2.8(D4×C30), (C4×C12)⋊12C10, C22⋊C4⋊5C30, (Q8×C30)⋊23C2, (Q8×C10)⋊13C6, (C2×D4).5C30, C6.71(D4×C10), C20.39(C3×D4), C10.71(C6×D4), C12.39(C5×D4), (D4×C30).26C2, (C6×D4).12C10, (D4×C10).12C6, C30.454(C2×D4), C23.3(C2×C30), C30.280(C4○D4), (C2×C60).469C22, (C2×C30).459C23, (C22×C30).3C22, C22.14(C22×C30), C6.44(C5×C4○D4), C2.7(C15×C4○D4), (C5×C22⋊C4)⋊13C6, (C2×C20).82(C2×C6), (C2×C4).13(C2×C30), C10.44(C3×C4○D4), (C15×C22⋊C4)⋊29C2, (C3×C22⋊C4)⋊13C10, (C2×C12).82(C2×C10), (C22×C10).7(C2×C6), (C22×C6).2(C2×C10), (C2×C10).79(C22×C6), (C2×C6).79(C22×C10), SmallGroup(480,929)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C4.4D4
G = < a,b,c,d | a15=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >
Subgroups: 232 in 152 conjugacy classes, 88 normal (32 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C10, C12, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C30, C30, C30, C4.4D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C4×C12, C3×C22⋊C4, C6×D4, C6×Q8, C60, C60, C2×C30, C2×C30, C4×C20, C5×C22⋊C4, D4×C10, Q8×C10, C3×C4.4D4, C2×C60, C2×C60, D4×C15, Q8×C15, C22×C30, C5×C4.4D4, C4×C60, C15×C22⋊C4, D4×C30, Q8×C30, C15×C4.4D4
Quotients: C1, C2, C3, C22, C5, C6, D4, C23, C10, C2×C6, C15, C2×D4, C4○D4, C2×C10, C3×D4, C22×C6, C30, C4.4D4, C5×D4, C22×C10, C6×D4, C3×C4○D4, C2×C30, D4×C10, C5×C4○D4, C3×C4.4D4, D4×C15, C22×C30, C5×C4.4D4, D4×C30, C15×C4○D4, C15×C4.4D4
►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 32 235 215)(2 33 236 216)(3 34 237 217)(4 35 238 218)(5 36 239 219)(6 37 240 220)(7 38 226 221)(8 39 227 222)(9 40 228 223)(10 41 229 224)(11 42 230 225)(12 43 231 211)(13 44 232 212)(14 45 233 213)(15 31 234 214)(16 158 74 80)(17 159 75 81)(18 160 61 82)(19 161 62 83)(20 162 63 84)(21 163 64 85)(22 164 65 86)(23 165 66 87)(24 151 67 88)(25 152 68 89)(26 153 69 90)(27 154 70 76)(28 155 71 77)(29 156 72 78)(30 157 73 79)(46 188 129 93)(47 189 130 94)(48 190 131 95)(49 191 132 96)(50 192 133 97)(51 193 134 98)(52 194 135 99)(53 195 121 100)(54 181 122 101)(55 182 123 102)(56 183 124 103)(57 184 125 104)(58 185 126 105)(59 186 127 91)(60 187 128 92)(106 172 138 199)(107 173 139 200)(108 174 140 201)(109 175 141 202)(110 176 142 203)(111 177 143 204)(112 178 144 205)(113 179 145 206)(114 180 146 207)(115 166 147 208)(116 167 148 209)(117 168 149 210)(118 169 150 196)(119 170 136 197)(120 171 137 198)
(1 159 100 136)(2 160 101 137)(3 161 102 138)(4 162 103 139)(5 163 104 140)(6 164 105 141)(7 165 91 142)(8 151 92 143)(9 152 93 144)(10 153 94 145)(11 154 95 146)(12 155 96 147)(13 156 97 148)(14 157 98 149)(15 158 99 150)(16 135 169 214)(17 121 170 215)(18 122 171 216)(19 123 172 217)(20 124 173 218)(21 125 174 219)(22 126 175 220)(23 127 176 221)(24 128 177 222)(25 129 178 223)(26 130 179 224)(27 131 180 225)(28 132 166 211)(29 133 167 212)(30 134 168 213)(31 74 52 196)(32 75 53 197)(33 61 54 198)(34 62 55 199)(35 63 56 200)(36 64 57 201)(37 65 58 202)(38 66 59 203)(39 67 60 204)(40 68 46 205)(41 69 47 206)(42 70 48 207)(43 71 49 208)(44 72 50 209)(45 73 51 210)(76 190 114 230)(77 191 115 231)(78 192 116 232)(79 193 117 233)(80 194 118 234)(81 195 119 235)(82 181 120 236)(83 182 106 237)(84 183 107 238)(85 184 108 239)(86 185 109 240)(87 186 110 226)(88 187 111 227)(89 188 112 228)(90 189 113 229)
(1 197 235 170)(2 198 236 171)(3 199 237 172)(4 200 238 173)(5 201 239 174)(6 202 240 175)(7 203 226 176)(8 204 227 177)(9 205 228 178)(10 206 229 179)(11 207 230 180)(12 208 231 166)(13 209 232 167)(14 210 233 168)(15 196 234 169)(16 99 74 194)(17 100 75 195)(18 101 61 181)(19 102 62 182)(20 103 63 183)(21 104 64 184)(22 105 65 185)(23 91 66 186)(24 92 67 187)(25 93 68 188)(26 94 69 189)(27 95 70 190)(28 96 71 191)(29 97 72 192)(30 98 73 193)(31 150 214 118)(32 136 215 119)(33 137 216 120)(34 138 217 106)(35 139 218 107)(36 140 219 108)(37 141 220 109)(38 142 221 110)(39 143 222 111)(40 144 223 112)(41 145 224 113)(42 146 225 114)(43 147 211 115)(44 148 212 116)(45 149 213 117)(46 152 129 89)(47 153 130 90)(48 154 131 76)(49 155 132 77)(50 156 133 78)(51 157 134 79)(52 158 135 80)(53 159 121 81)(54 160 122 82)(55 161 123 83)(56 162 124 84)(57 163 125 85)(58 164 126 86)(59 165 127 87)(60 151 128 88)
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), (1,32,235,215)(2,33,236,216)(3,34,237,217)(4,35,238,218)(5,36,239,219)(6,37,240,220)(7,38,226,221)(8,39,227,222)(9,40,228,223)(10,41,229,224)(11,42,230,225)(12,43,231,211)(13,44,232,212)(14,45,233,213)(15,31,234,214)(16,158,74,80)(17,159,75,81)(18,160,61,82)(19,161,62,83)(20,162,63,84)(21,163,64,85)(22,164,65,86)(23,165,66,87)(24,151,67,88)(25,152,68,89)(26,153,69,90)(27,154,70,76)(28,155,71,77)(29,156,72,78)(30,157,73,79)(46,188,129,93)(47,189,130,94)(48,190,131,95)(49,191,132,96)(50,192,133,97)(51,193,134,98)(52,194,135,99)(53,195,121,100)(54,181,122,101)(55,182,123,102)(56,183,124,103)(57,184,125,104)(58,185,126,105)(59,186,127,91)(60,187,128,92)(106,172,138,199)(107,173,139,200)(108,174,140,201)(109,175,141,202)(110,176,142,203)(111,177,143,204)(112,178,144,205)(113,179,145,206)(114,180,146,207)(115,166,147,208)(116,167,148,209)(117,168,149,210)(118,169,150,196)(119,170,136,197)(120,171,137,198), (1,159,100,136)(2,160,101,137)(3,161,102,138)(4,162,103,139)(5,163,104,140)(6,164,105,141)(7,165,91,142)(8,151,92,143)(9,152,93,144)(10,153,94,145)(11,154,95,146)(12,155,96,147)(13,156,97,148)(14,157,98,149)(15,158,99,150)(16,135,169,214)(17,121,170,215)(18,122,171,216)(19,123,172,217)(20,124,173,218)(21,125,174,219)(22,126,175,220)(23,127,176,221)(24,128,177,222)(25,129,178,223)(26,130,179,224)(27,131,180,225)(28,132,166,211)(29,133,167,212)(30,134,168,213)(31,74,52,196)(32,75,53,197)(33,61,54,198)(34,62,55,199)(35,63,56,200)(36,64,57,201)(37,65,58,202)(38,66,59,203)(39,67,60,204)(40,68,46,205)(41,69,47,206)(42,70,48,207)(43,71,49,208)(44,72,50,209)(45,73,51,210)(76,190,114,230)(77,191,115,231)(78,192,116,232)(79,193,117,233)(80,194,118,234)(81,195,119,235)(82,181,120,236)(83,182,106,237)(84,183,107,238)(85,184,108,239)(86,185,109,240)(87,186,110,226)(88,187,111,227)(89,188,112,228)(90,189,113,229), (1,197,235,170)(2,198,236,171)(3,199,237,172)(4,200,238,173)(5,201,239,174)(6,202,240,175)(7,203,226,176)(8,204,227,177)(9,205,228,178)(10,206,229,179)(11,207,230,180)(12,208,231,166)(13,209,232,167)(14,210,233,168)(15,196,234,169)(16,99,74,194)(17,100,75,195)(18,101,61,181)(19,102,62,182)(20,103,63,183)(21,104,64,184)(22,105,65,185)(23,91,66,186)(24,92,67,187)(25,93,68,188)(26,94,69,189)(27,95,70,190)(28,96,71,191)(29,97,72,192)(30,98,73,193)(31,150,214,118)(32,136,215,119)(33,137,216,120)(34,138,217,106)(35,139,218,107)(36,140,219,108)(37,141,220,109)(38,142,221,110)(39,143,222,111)(40,144,223,112)(41,145,224,113)(42,146,225,114)(43,147,211,115)(44,148,212,116)(45,149,213,117)(46,152,129,89)(47,153,130,90)(48,154,131,76)(49,155,132,77)(50,156,133,78)(51,157,134,79)(52,158,135,80)(53,159,121,81)(54,160,122,82)(55,161,123,83)(56,162,124,84)(57,163,125,85)(58,164,126,86)(59,165,127,87)(60,151,128,88)>;
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), (1,32,235,215)(2,33,236,216)(3,34,237,217)(4,35,238,218)(5,36,239,219)(6,37,240,220)(7,38,226,221)(8,39,227,222)(9,40,228,223)(10,41,229,224)(11,42,230,225)(12,43,231,211)(13,44,232,212)(14,45,233,213)(15,31,234,214)(16,158,74,80)(17,159,75,81)(18,160,61,82)(19,161,62,83)(20,162,63,84)(21,163,64,85)(22,164,65,86)(23,165,66,87)(24,151,67,88)(25,152,68,89)(26,153,69,90)(27,154,70,76)(28,155,71,77)(29,156,72,78)(30,157,73,79)(46,188,129,93)(47,189,130,94)(48,190,131,95)(49,191,132,96)(50,192,133,97)(51,193,134,98)(52,194,135,99)(53,195,121,100)(54,181,122,101)(55,182,123,102)(56,183,124,103)(57,184,125,104)(58,185,126,105)(59,186,127,91)(60,187,128,92)(106,172,138,199)(107,173,139,200)(108,174,140,201)(109,175,141,202)(110,176,142,203)(111,177,143,204)(112,178,144,205)(113,179,145,206)(114,180,146,207)(115,166,147,208)(116,167,148,209)(117,168,149,210)(118,169,150,196)(119,170,136,197)(120,171,137,198), (1,159,100,136)(2,160,101,137)(3,161,102,138)(4,162,103,139)(5,163,104,140)(6,164,105,141)(7,165,91,142)(8,151,92,143)(9,152,93,144)(10,153,94,145)(11,154,95,146)(12,155,96,147)(13,156,97,148)(14,157,98,149)(15,158,99,150)(16,135,169,214)(17,121,170,215)(18,122,171,216)(19,123,172,217)(20,124,173,218)(21,125,174,219)(22,126,175,220)(23,127,176,221)(24,128,177,222)(25,129,178,223)(26,130,179,224)(27,131,180,225)(28,132,166,211)(29,133,167,212)(30,134,168,213)(31,74,52,196)(32,75,53,197)(33,61,54,198)(34,62,55,199)(35,63,56,200)(36,64,57,201)(37,65,58,202)(38,66,59,203)(39,67,60,204)(40,68,46,205)(41,69,47,206)(42,70,48,207)(43,71,49,208)(44,72,50,209)(45,73,51,210)(76,190,114,230)(77,191,115,231)(78,192,116,232)(79,193,117,233)(80,194,118,234)(81,195,119,235)(82,181,120,236)(83,182,106,237)(84,183,107,238)(85,184,108,239)(86,185,109,240)(87,186,110,226)(88,187,111,227)(89,188,112,228)(90,189,113,229), (1,197,235,170)(2,198,236,171)(3,199,237,172)(4,200,238,173)(5,201,239,174)(6,202,240,175)(7,203,226,176)(8,204,227,177)(9,205,228,178)(10,206,229,179)(11,207,230,180)(12,208,231,166)(13,209,232,167)(14,210,233,168)(15,196,234,169)(16,99,74,194)(17,100,75,195)(18,101,61,181)(19,102,62,182)(20,103,63,183)(21,104,64,184)(22,105,65,185)(23,91,66,186)(24,92,67,187)(25,93,68,188)(26,94,69,189)(27,95,70,190)(28,96,71,191)(29,97,72,192)(30,98,73,193)(31,150,214,118)(32,136,215,119)(33,137,216,120)(34,138,217,106)(35,139,218,107)(36,140,219,108)(37,141,220,109)(38,142,221,110)(39,143,222,111)(40,144,223,112)(41,145,224,113)(42,146,225,114)(43,147,211,115)(44,148,212,116)(45,149,213,117)(46,152,129,89)(47,153,130,90)(48,154,131,76)(49,155,132,77)(50,156,133,78)(51,157,134,79)(52,158,135,80)(53,159,121,81)(54,160,122,82)(55,161,123,83)(56,162,124,84)(57,163,125,85)(58,164,126,86)(59,165,127,87)(60,151,128,88) );
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)], [(1,32,235,215),(2,33,236,216),(3,34,237,217),(4,35,238,218),(5,36,239,219),(6,37,240,220),(7,38,226,221),(8,39,227,222),(9,40,228,223),(10,41,229,224),(11,42,230,225),(12,43,231,211),(13,44,232,212),(14,45,233,213),(15,31,234,214),(16,158,74,80),(17,159,75,81),(18,160,61,82),(19,161,62,83),(20,162,63,84),(21,163,64,85),(22,164,65,86),(23,165,66,87),(24,151,67,88),(25,152,68,89),(26,153,69,90),(27,154,70,76),(28,155,71,77),(29,156,72,78),(30,157,73,79),(46,188,129,93),(47,189,130,94),(48,190,131,95),(49,191,132,96),(50,192,133,97),(51,193,134,98),(52,194,135,99),(53,195,121,100),(54,181,122,101),(55,182,123,102),(56,183,124,103),(57,184,125,104),(58,185,126,105),(59,186,127,91),(60,187,128,92),(106,172,138,199),(107,173,139,200),(108,174,140,201),(109,175,141,202),(110,176,142,203),(111,177,143,204),(112,178,144,205),(113,179,145,206),(114,180,146,207),(115,166,147,208),(116,167,148,209),(117,168,149,210),(118,169,150,196),(119,170,136,197),(120,171,137,198)], [(1,159,100,136),(2,160,101,137),(3,161,102,138),(4,162,103,139),(5,163,104,140),(6,164,105,141),(7,165,91,142),(8,151,92,143),(9,152,93,144),(10,153,94,145),(11,154,95,146),(12,155,96,147),(13,156,97,148),(14,157,98,149),(15,158,99,150),(16,135,169,214),(17,121,170,215),(18,122,171,216),(19,123,172,217),(20,124,173,218),(21,125,174,219),(22,126,175,220),(23,127,176,221),(24,128,177,222),(25,129,178,223),(26,130,179,224),(27,131,180,225),(28,132,166,211),(29,133,167,212),(30,134,168,213),(31,74,52,196),(32,75,53,197),(33,61,54,198),(34,62,55,199),(35,63,56,200),(36,64,57,201),(37,65,58,202),(38,66,59,203),(39,67,60,204),(40,68,46,205),(41,69,47,206),(42,70,48,207),(43,71,49,208),(44,72,50,209),(45,73,51,210),(76,190,114,230),(77,191,115,231),(78,192,116,232),(79,193,117,233),(80,194,118,234),(81,195,119,235),(82,181,120,236),(83,182,106,237),(84,183,107,238),(85,184,108,239),(86,185,109,240),(87,186,110,226),(88,187,111,227),(89,188,112,228),(90,189,113,229)], [(1,197,235,170),(2,198,236,171),(3,199,237,172),(4,200,238,173),(5,201,239,174),(6,202,240,175),(7,203,226,176),(8,204,227,177),(9,205,228,178),(10,206,229,179),(11,207,230,180),(12,208,231,166),(13,209,232,167),(14,210,233,168),(15,196,234,169),(16,99,74,194),(17,100,75,195),(18,101,61,181),(19,102,62,182),(20,103,63,183),(21,104,64,184),(22,105,65,185),(23,91,66,186),(24,92,67,187),(25,93,68,188),(26,94,69,189),(27,95,70,190),(28,96,71,191),(29,97,72,192),(30,98,73,193),(31,150,214,118),(32,136,215,119),(33,137,216,120),(34,138,217,106),(35,139,218,107),(36,140,219,108),(37,141,220,109),(38,142,221,110),(39,143,222,111),(40,144,223,112),(41,145,224,113),(42,146,225,114),(43,147,211,115),(44,148,212,116),(45,149,213,117),(46,152,129,89),(47,153,130,90),(48,154,131,76),(49,155,132,77),(50,156,133,78),(51,157,134,79),(52,158,135,80),(53,159,121,81),(54,160,122,82),(55,161,123,83),(56,162,124,84),(57,163,125,85),(58,164,126,86),(59,165,127,87),(60,151,128,88)]])
210 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 10A | ··· | 10L | 10M | ··· | 10T | 12A | ··· | 12L | 12M | 12N | 12O | 12P | 15A | ··· | 15H | 20A | ··· | 20X | 20Y | ··· | 20AF | 30A | ··· | 30X | 30Y | ··· | 30AN | 60A | ··· | 60AV | 60AW | ··· | 60BL |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 2 | ··· | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
210 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C5 | C6 | C6 | C6 | C6 | C10 | C10 | C10 | C10 | C15 | C30 | C30 | C30 | C30 | D4 | C4○D4 | C3×D4 | C5×D4 | C3×C4○D4 | C5×C4○D4 | D4×C15 | C15×C4○D4 |
| kernel | C15×C4.4D4 | C4×C60 | C15×C22⋊C4 | D4×C30 | Q8×C30 | C5×C4.4D4 | C3×C4.4D4 | C4×C20 | C5×C22⋊C4 | D4×C10 | Q8×C10 | C4×C12 | C3×C22⋊C4 | C6×D4 | C6×Q8 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C60 | C30 | C20 | C12 | C10 | C6 | C4 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 2 | 4 | 2 | 8 | 2 | 2 | 4 | 16 | 4 | 4 | 8 | 8 | 32 | 8 | 8 | 2 | 4 | 4 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C15×C4.4D4 ►in GL4(𝔽61) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 37 | 32 | 0 | 0 |
| 22 | 24 | 0 | 0 |
| 0 | 0 | 10 | 22 |
| 0 | 0 | 37 | 51 |
| 41 | 47 | 0 | 0 |
| 59 | 20 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 11 |
| 50 | 37 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 49 | 59 |
| 0 | 0 | 42 | 12 |
G:=sub<GL(4,GF(61))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[37,22,0,0,32,24,0,0,0,0,10,37,0,0,22,51],[41,59,0,0,47,20,0,0,0,0,11,0,0,0,0,11],[50,0,0,0,37,11,0,0,0,0,49,42,0,0,59,12] >;
C15×C4.4D4 in GAP, Magma, Sage, TeX
C_{15}\times C_4._4D_4 % in TeX
G:=Group("C15xC4.4D4"); // GroupNames label
G:=SmallGroup(480,929);
// by ID
G=gap.SmallGroup(480,929);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,-2,1709,1688,5126,646]);
// Polycyclic
G:=Group<a,b,c,d|a^15=b^4=c^4=1,d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^2*c^-1>;
// generators/relations