direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C15×C8.C4, C8.1C60, C24.2C20, C120.23C4, C40.10C12, C60.253D4, M4(2).2C30, C4.8(C2×C60), (C2×C8).5C30, (C2×C40).15C6, C22.(Q8×C15), C20.68(C3×D4), C12.68(C5×D4), C4.19(D4×C15), C30.66(C4⋊C4), (C2×C30).11Q8, C60.259(C2×C4), (C2×C120).31C2, C20.66(C2×C12), C12.45(C2×C20), (C2×C24).11C10, (C5×M4(2)).4C6, (C2×C60).574C22, (C3×M4(2)).4C10, (C15×M4(2)).8C2, C2.5(C15×C4⋊C4), C6.14(C5×C4⋊C4), C10.21(C3×C4⋊C4), (C2×C6).2(C5×Q8), (C2×C10).2(C3×Q8), (C2×C4).21(C2×C30), (C2×C20).122(C2×C6), (C2×C12).125(C2×C10), SmallGroup(480,211)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C15×C8.C4
G = < a,b,c | a15=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 72 in 60 conjugacy classes, 48 normal (40 characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C8, C8, C2×C4, C10, C10, C12, C2×C6, C15, C2×C8, M4(2), C20, C2×C10, C24, C24, C2×C12, C30, C30, C8.C4, C40, C40, C2×C20, C2×C24, C3×M4(2), C60, C2×C30, C2×C40, C5×M4(2), C3×C8.C4, C120, C120, C2×C60, C5×C8.C4, C2×C120, C15×M4(2), C15×C8.C4
Quotients: C1, C2, C3, C4, C22, C5, C6, C2×C4, D4, Q8, C10, C12, C2×C6, C15, C4⋊C4, C20, C2×C10, C2×C12, C3×D4, C3×Q8, C30, C8.C4, C2×C20, C5×D4, C5×Q8, C3×C4⋊C4, C60, C2×C30, C5×C4⋊C4, C3×C8.C4, C2×C60, D4×C15, Q8×C15, C5×C8.C4, C15×C4⋊C4, C15×C8.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 98 226 77 157 173 130 54)(2 99 227 78 158 174 131 55)(3 100 228 79 159 175 132 56)(4 101 229 80 160 176 133 57)(5 102 230 81 161 177 134 58)(6 103 231 82 162 178 135 59)(7 104 232 83 163 179 121 60)(8 105 233 84 164 180 122 46)(9 91 234 85 165 166 123 47)(10 92 235 86 151 167 124 48)(11 93 236 87 152 168 125 49)(12 94 237 88 153 169 126 50)(13 95 238 89 154 170 127 51)(14 96 239 90 155 171 128 52)(15 97 240 76 156 172 129 53)(16 217 144 203 64 186 114 33)(17 218 145 204 65 187 115 34)(18 219 146 205 66 188 116 35)(19 220 147 206 67 189 117 36)(20 221 148 207 68 190 118 37)(21 222 149 208 69 191 119 38)(22 223 150 209 70 192 120 39)(23 224 136 210 71 193 106 40)(24 225 137 196 72 194 107 41)(25 211 138 197 73 195 108 42)(26 212 139 198 74 181 109 43)(27 213 140 199 75 182 110 44)(28 214 141 200 61 183 111 45)(29 215 142 201 62 184 112 31)(30 216 143 202 63 185 113 32)
(1 35 226 188 157 205 130 219)(2 36 227 189 158 206 131 220)(3 37 228 190 159 207 132 221)(4 38 229 191 160 208 133 222)(5 39 230 192 161 209 134 223)(6 40 231 193 162 210 135 224)(7 41 232 194 163 196 121 225)(8 42 233 195 164 197 122 211)(9 43 234 181 165 198 123 212)(10 44 235 182 151 199 124 213)(11 45 236 183 152 200 125 214)(12 31 237 184 153 201 126 215)(13 32 238 185 154 202 127 216)(14 33 239 186 155 203 128 217)(15 34 240 187 156 204 129 218)(16 96 114 90 64 171 144 52)(17 97 115 76 65 172 145 53)(18 98 116 77 66 173 146 54)(19 99 117 78 67 174 147 55)(20 100 118 79 68 175 148 56)(21 101 119 80 69 176 149 57)(22 102 120 81 70 177 150 58)(23 103 106 82 71 178 136 59)(24 104 107 83 72 179 137 60)(25 105 108 84 73 180 138 46)(26 91 109 85 74 166 139 47)(27 92 110 86 75 167 140 48)(28 93 111 87 61 168 141 49)(29 94 112 88 62 169 142 50)(30 95 113 89 63 170 143 51)
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), (1,98,226,77,157,173,130,54)(2,99,227,78,158,174,131,55)(3,100,228,79,159,175,132,56)(4,101,229,80,160,176,133,57)(5,102,230,81,161,177,134,58)(6,103,231,82,162,178,135,59)(7,104,232,83,163,179,121,60)(8,105,233,84,164,180,122,46)(9,91,234,85,165,166,123,47)(10,92,235,86,151,167,124,48)(11,93,236,87,152,168,125,49)(12,94,237,88,153,169,126,50)(13,95,238,89,154,170,127,51)(14,96,239,90,155,171,128,52)(15,97,240,76,156,172,129,53)(16,217,144,203,64,186,114,33)(17,218,145,204,65,187,115,34)(18,219,146,205,66,188,116,35)(19,220,147,206,67,189,117,36)(20,221,148,207,68,190,118,37)(21,222,149,208,69,191,119,38)(22,223,150,209,70,192,120,39)(23,224,136,210,71,193,106,40)(24,225,137,196,72,194,107,41)(25,211,138,197,73,195,108,42)(26,212,139,198,74,181,109,43)(27,213,140,199,75,182,110,44)(28,214,141,200,61,183,111,45)(29,215,142,201,62,184,112,31)(30,216,143,202,63,185,113,32), (1,35,226,188,157,205,130,219)(2,36,227,189,158,206,131,220)(3,37,228,190,159,207,132,221)(4,38,229,191,160,208,133,222)(5,39,230,192,161,209,134,223)(6,40,231,193,162,210,135,224)(7,41,232,194,163,196,121,225)(8,42,233,195,164,197,122,211)(9,43,234,181,165,198,123,212)(10,44,235,182,151,199,124,213)(11,45,236,183,152,200,125,214)(12,31,237,184,153,201,126,215)(13,32,238,185,154,202,127,216)(14,33,239,186,155,203,128,217)(15,34,240,187,156,204,129,218)(16,96,114,90,64,171,144,52)(17,97,115,76,65,172,145,53)(18,98,116,77,66,173,146,54)(19,99,117,78,67,174,147,55)(20,100,118,79,68,175,148,56)(21,101,119,80,69,176,149,57)(22,102,120,81,70,177,150,58)(23,103,106,82,71,178,136,59)(24,104,107,83,72,179,137,60)(25,105,108,84,73,180,138,46)(26,91,109,85,74,166,139,47)(27,92,110,86,75,167,140,48)(28,93,111,87,61,168,141,49)(29,94,112,88,62,169,142,50)(30,95,113,89,63,170,143,51)>;
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), (1,98,226,77,157,173,130,54)(2,99,227,78,158,174,131,55)(3,100,228,79,159,175,132,56)(4,101,229,80,160,176,133,57)(5,102,230,81,161,177,134,58)(6,103,231,82,162,178,135,59)(7,104,232,83,163,179,121,60)(8,105,233,84,164,180,122,46)(9,91,234,85,165,166,123,47)(10,92,235,86,151,167,124,48)(11,93,236,87,152,168,125,49)(12,94,237,88,153,169,126,50)(13,95,238,89,154,170,127,51)(14,96,239,90,155,171,128,52)(15,97,240,76,156,172,129,53)(16,217,144,203,64,186,114,33)(17,218,145,204,65,187,115,34)(18,219,146,205,66,188,116,35)(19,220,147,206,67,189,117,36)(20,221,148,207,68,190,118,37)(21,222,149,208,69,191,119,38)(22,223,150,209,70,192,120,39)(23,224,136,210,71,193,106,40)(24,225,137,196,72,194,107,41)(25,211,138,197,73,195,108,42)(26,212,139,198,74,181,109,43)(27,213,140,199,75,182,110,44)(28,214,141,200,61,183,111,45)(29,215,142,201,62,184,112,31)(30,216,143,202,63,185,113,32), (1,35,226,188,157,205,130,219)(2,36,227,189,158,206,131,220)(3,37,228,190,159,207,132,221)(4,38,229,191,160,208,133,222)(5,39,230,192,161,209,134,223)(6,40,231,193,162,210,135,224)(7,41,232,194,163,196,121,225)(8,42,233,195,164,197,122,211)(9,43,234,181,165,198,123,212)(10,44,235,182,151,199,124,213)(11,45,236,183,152,200,125,214)(12,31,237,184,153,201,126,215)(13,32,238,185,154,202,127,216)(14,33,239,186,155,203,128,217)(15,34,240,187,156,204,129,218)(16,96,114,90,64,171,144,52)(17,97,115,76,65,172,145,53)(18,98,116,77,66,173,146,54)(19,99,117,78,67,174,147,55)(20,100,118,79,68,175,148,56)(21,101,119,80,69,176,149,57)(22,102,120,81,70,177,150,58)(23,103,106,82,71,178,136,59)(24,104,107,83,72,179,137,60)(25,105,108,84,73,180,138,46)(26,91,109,85,74,166,139,47)(27,92,110,86,75,167,140,48)(28,93,111,87,61,168,141,49)(29,94,112,88,62,169,142,50)(30,95,113,89,63,170,143,51) );
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)], [(1,98,226,77,157,173,130,54),(2,99,227,78,158,174,131,55),(3,100,228,79,159,175,132,56),(4,101,229,80,160,176,133,57),(5,102,230,81,161,177,134,58),(6,103,231,82,162,178,135,59),(7,104,232,83,163,179,121,60),(8,105,233,84,164,180,122,46),(9,91,234,85,165,166,123,47),(10,92,235,86,151,167,124,48),(11,93,236,87,152,168,125,49),(12,94,237,88,153,169,126,50),(13,95,238,89,154,170,127,51),(14,96,239,90,155,171,128,52),(15,97,240,76,156,172,129,53),(16,217,144,203,64,186,114,33),(17,218,145,204,65,187,115,34),(18,219,146,205,66,188,116,35),(19,220,147,206,67,189,117,36),(20,221,148,207,68,190,118,37),(21,222,149,208,69,191,119,38),(22,223,150,209,70,192,120,39),(23,224,136,210,71,193,106,40),(24,225,137,196,72,194,107,41),(25,211,138,197,73,195,108,42),(26,212,139,198,74,181,109,43),(27,213,140,199,75,182,110,44),(28,214,141,200,61,183,111,45),(29,215,142,201,62,184,112,31),(30,216,143,202,63,185,113,32)], [(1,35,226,188,157,205,130,219),(2,36,227,189,158,206,131,220),(3,37,228,190,159,207,132,221),(4,38,229,191,160,208,133,222),(5,39,230,192,161,209,134,223),(6,40,231,193,162,210,135,224),(7,41,232,194,163,196,121,225),(8,42,233,195,164,197,122,211),(9,43,234,181,165,198,123,212),(10,44,235,182,151,199,124,213),(11,45,236,183,152,200,125,214),(12,31,237,184,153,201,126,215),(13,32,238,185,154,202,127,216),(14,33,239,186,155,203,128,217),(15,34,240,187,156,204,129,218),(16,96,114,90,64,171,144,52),(17,97,115,76,65,172,145,53),(18,98,116,77,66,173,146,54),(19,99,117,78,67,174,147,55),(20,100,118,79,68,175,148,56),(21,101,119,80,69,176,149,57),(22,102,120,81,70,177,150,58),(23,103,106,82,71,178,136,59),(24,104,107,83,72,179,137,60),(25,105,108,84,73,180,138,46),(26,91,109,85,74,166,139,47),(27,92,110,86,75,167,140,48),(28,93,111,87,61,168,141,49),(29,94,112,88,62,169,142,50),(30,95,113,89,63,170,143,51)]])
210 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 12E | 12F | 15A | ··· | 15H | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 24A | ··· | 24H | 24I | ··· | 24P | 30A | ··· | 30H | 30I | ··· | 30P | 40A | ··· | 40P | 40Q | ··· | 40AF | 60A | ··· | 60P | 60Q | ··· | 60X | 120A | ··· | 120AF | 120AG | ··· | 120BL |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | ··· | 15 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 24 | ··· | 24 | 30 | ··· | 30 | 30 | ··· | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | ··· | 60 | 120 | ··· | 120 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
210 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C5 | C6 | C6 | C10 | C10 | C12 | C15 | C20 | C30 | C30 | C60 | D4 | Q8 | C3×D4 | C3×Q8 | C8.C4 | C5×D4 | C5×Q8 | C3×C8.C4 | D4×C15 | Q8×C15 | C5×C8.C4 | C15×C8.C4 |
| kernel | C15×C8.C4 | C2×C120 | C15×M4(2) | C5×C8.C4 | C120 | C3×C8.C4 | C2×C40 | C5×M4(2) | C2×C24 | C3×M4(2) | C40 | C8.C4 | C24 | C2×C8 | M4(2) | C8 | C60 | C2×C30 | C20 | C2×C10 | C15 | C12 | C2×C6 | C5 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 8 | 8 | 8 | 16 | 8 | 16 | 32 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 16 | 32 |
Matrix representation of C15×C8.C4 ►in GL3(𝔽241) generated by
| 15 | 0 | 0 |
| 0 | 91 | 0 |
| 0 | 0 | 91 |
| 240 | 0 | 0 |
| 0 | 211 | 129 |
| 0 | 0 | 8 |
| 177 | 0 | 0 |
| 0 | 54 | 165 |
| 0 | 128 | 187 |
G:=sub<GL(3,GF(241))| [15,0,0,0,91,0,0,0,91],[240,0,0,0,211,0,0,129,8],[177,0,0,0,54,128,0,165,187] >;
C15×C8.C4 in GAP, Magma, Sage, TeX
C_{15}\times C_8.C_4 % in TeX
G:=Group("C15xC8.C4"); // GroupNames label
G:=SmallGroup(480,211);
// by ID
G=gap.SmallGroup(480,211);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,-2,-2,840,869,428,10504,172,124]);
// Polycyclic
G:=Group<a,b,c|a^15=b^8=1,c^4=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations