metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.55D6, C24.37D10, C60.172C23, C120.59C22, (S3×C8)⋊8D5, (S3×C40)⋊9C2, C15⋊Q8.4C4, C8⋊D5⋊7S3, C15⋊9(C8○D4), C3⋊C8.36D10, D6.4(C4×D5), C8.15(S3×D5), C5⋊4(D12.C4), (C4×D5).56D6, D10.5(C4×S3), C5⋊2C8.23D6, C5⋊D12.4C4, C15⋊D4.4C4, C3⋊D20.4C4, C40⋊S3⋊12C2, D30.19(C2×C4), (C4×S3).44D10, Dic3.6(C4×D5), Dic5.5(C4×S3), D15⋊2C8⋊10C2, C3⋊2(D20.3C4), D6.Dic5⋊11C2, C30.38(C22×C4), (S3×C20).52C22, C20.169(C22×S3), C15⋊3C8.28C22, Dic15.20(C2×C4), D6.D10.2C2, (D5×C12).56C22, (C4×D15).39C22, C12.169(C22×D5), (D5×C3⋊C8)⋊9C2, C6.7(C2×C4×D5), C2.10(C4×S3×D5), C10.38(S3×C2×C4), C4.142(C2×S3×D5), (C3×C8⋊D5)⋊9C2, (C6×D5).3(C2×C4), (C5×C3⋊C8).38C22, (S3×C10).27(C2×C4), (C3×Dic5).3(C2×C4), (C3×C5⋊2C8).23C22, (C5×Dic3).32(C2×C4), SmallGroup(480,343)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40.55D6
G = < a,b,c | a40=b6=1, c2=a20, bab-1=cac-1=a29, cbc-1=a20b-1 >
Subgroups: 540 in 124 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C8, C2×C4, D4, Q8, D5, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C15, C2×C8, M4(2), C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C3⋊C8, C3⋊C8, C24, C24, Dic6, C4×S3, C4×S3, D12, C3⋊D4, C2×C12, C5×S3, C3×D5, D15, C30, C8○D4, C5⋊2C8, C5⋊2C8, C40, C40, Dic10, C4×D5, C4×D5, D20, C5⋊D4, C2×C20, S3×C8, S3×C8, C8⋊S3, C2×C3⋊C8, C3×M4(2), C4○D12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, S3×C10, D30, C8×D5, C8⋊D5, C8⋊D5, C4.Dic5, C2×C40, C4○D20, D12.C4, C5×C3⋊C8, C3×C5⋊2C8, C15⋊3C8, C120, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, S3×C20, C4×D15, D20.3C4, D5×C3⋊C8, D15⋊2C8, D6.Dic5, C3×C8⋊D5, S3×C40, C40⋊S3, D6.D10, C40.55D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D5, D6, C22×C4, D10, C4×S3, C22×S3, C8○D4, C4×D5, C22×D5, S3×C2×C4, S3×D5, C2×C4×D5, D12.C4, C2×S3×D5, D20.3C4, C4×S3×D5, C40.55D6
►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)
(1 132 108)(2 121 109 30 133 97)(3 150 110 19 134 86)(4 139 111 8 135 115)(5 128 112 37 136 104)(6 157 113 26 137 93)(7 146 114 15 138 82)(9 124 116 33 140 100)(10 153 117 22 141 89)(11 142 118)(12 131 119 40 143 107)(13 160 120 29 144 96)(14 149 81 18 145 85)(16 127 83 36 147 103)(17 156 84 25 148 92)(20 123 87 32 151 99)(21 152 88)(23 130 90 39 154 106)(24 159 91 28 155 95)(27 126 94 35 158 102)(31 122 98)(34 129 101 38 125 105)(41 195 211 57 179 227)(42 184 212 46 180 216)(43 173 213 75 181 205)(44 162 214 64 182 234)(45 191 215 53 183 223)(47 169 217 71 185 201)(48 198 218 60 186 230)(49 187 219)(50 176 220 78 188 208)(51 165 221 67 189 237)(52 194 222 56 190 226)(54 172 224 74 192 204)(55 161 225 63 193 233)(58 168 228 70 196 240)(59 197 229)(61 175 231 77 199 207)(62 164 232 66 200 236)(65 171 235 73 163 203)(68 178 238 80 166 210)(69 167 239)(72 174 202 76 170 206)(79 177 209)
(1 54 21 74)(2 43 22 63)(3 72 23 52)(4 61 24 41)(5 50 25 70)(6 79 26 59)(7 68 27 48)(8 57 28 77)(9 46 29 66)(10 75 30 55)(11 64 31 44)(12 53 32 73)(13 42 33 62)(14 71 34 51)(15 60 35 80)(16 49 36 69)(17 78 37 58)(18 67 38 47)(19 56 39 76)(20 45 40 65)(81 169 101 189)(82 198 102 178)(83 187 103 167)(84 176 104 196)(85 165 105 185)(86 194 106 174)(87 183 107 163)(88 172 108 192)(89 161 109 181)(90 190 110 170)(91 179 111 199)(92 168 112 188)(93 197 113 177)(94 186 114 166)(95 175 115 195)(96 164 116 184)(97 193 117 173)(98 182 118 162)(99 171 119 191)(100 200 120 180)(121 225 141 205)(122 214 142 234)(123 203 143 223)(124 232 144 212)(125 221 145 201)(126 210 146 230)(127 239 147 219)(128 228 148 208)(129 217 149 237)(130 206 150 226)(131 235 151 215)(132 224 152 204)(133 213 153 233)(134 202 154 222)(135 231 155 211)(136 220 156 240)(137 209 157 229)(138 238 158 218)(139 227 159 207)(140 216 160 236)
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), (1,132,108)(2,121,109,30,133,97)(3,150,110,19,134,86)(4,139,111,8,135,115)(5,128,112,37,136,104)(6,157,113,26,137,93)(7,146,114,15,138,82)(9,124,116,33,140,100)(10,153,117,22,141,89)(11,142,118)(12,131,119,40,143,107)(13,160,120,29,144,96)(14,149,81,18,145,85)(16,127,83,36,147,103)(17,156,84,25,148,92)(20,123,87,32,151,99)(21,152,88)(23,130,90,39,154,106)(24,159,91,28,155,95)(27,126,94,35,158,102)(31,122,98)(34,129,101,38,125,105)(41,195,211,57,179,227)(42,184,212,46,180,216)(43,173,213,75,181,205)(44,162,214,64,182,234)(45,191,215,53,183,223)(47,169,217,71,185,201)(48,198,218,60,186,230)(49,187,219)(50,176,220,78,188,208)(51,165,221,67,189,237)(52,194,222,56,190,226)(54,172,224,74,192,204)(55,161,225,63,193,233)(58,168,228,70,196,240)(59,197,229)(61,175,231,77,199,207)(62,164,232,66,200,236)(65,171,235,73,163,203)(68,178,238,80,166,210)(69,167,239)(72,174,202,76,170,206)(79,177,209), (1,54,21,74)(2,43,22,63)(3,72,23,52)(4,61,24,41)(5,50,25,70)(6,79,26,59)(7,68,27,48)(8,57,28,77)(9,46,29,66)(10,75,30,55)(11,64,31,44)(12,53,32,73)(13,42,33,62)(14,71,34,51)(15,60,35,80)(16,49,36,69)(17,78,37,58)(18,67,38,47)(19,56,39,76)(20,45,40,65)(81,169,101,189)(82,198,102,178)(83,187,103,167)(84,176,104,196)(85,165,105,185)(86,194,106,174)(87,183,107,163)(88,172,108,192)(89,161,109,181)(90,190,110,170)(91,179,111,199)(92,168,112,188)(93,197,113,177)(94,186,114,166)(95,175,115,195)(96,164,116,184)(97,193,117,173)(98,182,118,162)(99,171,119,191)(100,200,120,180)(121,225,141,205)(122,214,142,234)(123,203,143,223)(124,232,144,212)(125,221,145,201)(126,210,146,230)(127,239,147,219)(128,228,148,208)(129,217,149,237)(130,206,150,226)(131,235,151,215)(132,224,152,204)(133,213,153,233)(134,202,154,222)(135,231,155,211)(136,220,156,240)(137,209,157,229)(138,238,158,218)(139,227,159,207)(140,216,160,236)>;
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), (1,132,108)(2,121,109,30,133,97)(3,150,110,19,134,86)(4,139,111,8,135,115)(5,128,112,37,136,104)(6,157,113,26,137,93)(7,146,114,15,138,82)(9,124,116,33,140,100)(10,153,117,22,141,89)(11,142,118)(12,131,119,40,143,107)(13,160,120,29,144,96)(14,149,81,18,145,85)(16,127,83,36,147,103)(17,156,84,25,148,92)(20,123,87,32,151,99)(21,152,88)(23,130,90,39,154,106)(24,159,91,28,155,95)(27,126,94,35,158,102)(31,122,98)(34,129,101,38,125,105)(41,195,211,57,179,227)(42,184,212,46,180,216)(43,173,213,75,181,205)(44,162,214,64,182,234)(45,191,215,53,183,223)(47,169,217,71,185,201)(48,198,218,60,186,230)(49,187,219)(50,176,220,78,188,208)(51,165,221,67,189,237)(52,194,222,56,190,226)(54,172,224,74,192,204)(55,161,225,63,193,233)(58,168,228,70,196,240)(59,197,229)(61,175,231,77,199,207)(62,164,232,66,200,236)(65,171,235,73,163,203)(68,178,238,80,166,210)(69,167,239)(72,174,202,76,170,206)(79,177,209), (1,54,21,74)(2,43,22,63)(3,72,23,52)(4,61,24,41)(5,50,25,70)(6,79,26,59)(7,68,27,48)(8,57,28,77)(9,46,29,66)(10,75,30,55)(11,64,31,44)(12,53,32,73)(13,42,33,62)(14,71,34,51)(15,60,35,80)(16,49,36,69)(17,78,37,58)(18,67,38,47)(19,56,39,76)(20,45,40,65)(81,169,101,189)(82,198,102,178)(83,187,103,167)(84,176,104,196)(85,165,105,185)(86,194,106,174)(87,183,107,163)(88,172,108,192)(89,161,109,181)(90,190,110,170)(91,179,111,199)(92,168,112,188)(93,197,113,177)(94,186,114,166)(95,175,115,195)(96,164,116,184)(97,193,117,173)(98,182,118,162)(99,171,119,191)(100,200,120,180)(121,225,141,205)(122,214,142,234)(123,203,143,223)(124,232,144,212)(125,221,145,201)(126,210,146,230)(127,239,147,219)(128,228,148,208)(129,217,149,237)(130,206,150,226)(131,235,151,215)(132,224,152,204)(133,213,153,233)(134,202,154,222)(135,231,155,211)(136,220,156,240)(137,209,157,229)(138,238,158,218)(139,227,159,207)(140,216,160,236) );
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)], [(1,132,108),(2,121,109,30,133,97),(3,150,110,19,134,86),(4,139,111,8,135,115),(5,128,112,37,136,104),(6,157,113,26,137,93),(7,146,114,15,138,82),(9,124,116,33,140,100),(10,153,117,22,141,89),(11,142,118),(12,131,119,40,143,107),(13,160,120,29,144,96),(14,149,81,18,145,85),(16,127,83,36,147,103),(17,156,84,25,148,92),(20,123,87,32,151,99),(21,152,88),(23,130,90,39,154,106),(24,159,91,28,155,95),(27,126,94,35,158,102),(31,122,98),(34,129,101,38,125,105),(41,195,211,57,179,227),(42,184,212,46,180,216),(43,173,213,75,181,205),(44,162,214,64,182,234),(45,191,215,53,183,223),(47,169,217,71,185,201),(48,198,218,60,186,230),(49,187,219),(50,176,220,78,188,208),(51,165,221,67,189,237),(52,194,222,56,190,226),(54,172,224,74,192,204),(55,161,225,63,193,233),(58,168,228,70,196,240),(59,197,229),(61,175,231,77,199,207),(62,164,232,66,200,236),(65,171,235,73,163,203),(68,178,238,80,166,210),(69,167,239),(72,174,202,76,170,206),(79,177,209)], [(1,54,21,74),(2,43,22,63),(3,72,23,52),(4,61,24,41),(5,50,25,70),(6,79,26,59),(7,68,27,48),(8,57,28,77),(9,46,29,66),(10,75,30,55),(11,64,31,44),(12,53,32,73),(13,42,33,62),(14,71,34,51),(15,60,35,80),(16,49,36,69),(17,78,37,58),(18,67,38,47),(19,56,39,76),(20,45,40,65),(81,169,101,189),(82,198,102,178),(83,187,103,167),(84,176,104,196),(85,165,105,185),(86,194,106,174),(87,183,107,163),(88,172,108,192),(89,161,109,181),(90,190,110,170),(91,179,111,199),(92,168,112,188),(93,197,113,177),(94,186,114,166),(95,175,115,195),(96,164,116,184),(97,193,117,173),(98,182,118,162),(99,171,119,191),(100,200,120,180),(121,225,141,205),(122,214,142,234),(123,203,143,223),(124,232,144,212),(125,221,145,201),(126,210,146,230),(127,239,147,219),(128,228,148,208),(129,217,149,237),(130,206,150,226),(131,235,151,215),(132,224,152,204),(133,213,153,233),(134,202,154,222),(135,231,155,211),(136,220,156,240),(137,209,157,229),(138,238,158,218),(139,227,159,207),(140,216,160,236)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 24A | 24B | 24C | 24D | 30A | 30B | 40A | ··· | 40H | 40I | ··· | 40P | 60A | 60B | 60C | 60D | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | ··· | 40 | 40 | ··· | 40 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 6 | 10 | 30 | 2 | 1 | 1 | 6 | 10 | 30 | 2 | 2 | 2 | 20 | 2 | 2 | 3 | 3 | 3 | 3 | 10 | 10 | 30 | 30 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C4×S3 | C4×S3 | C8○D4 | C4×D5 | C4×D5 | D20.3C4 | S3×D5 | D12.C4 | C2×S3×D5 | C4×S3×D5 | C40.55D6 |
| kernel | C40.55D6 | D5×C3⋊C8 | D15⋊2C8 | D6.Dic5 | C3×C8⋊D5 | S3×C40 | C40⋊S3 | D6.D10 | C15⋊D4 | C3⋊D20 | C5⋊D12 | C15⋊Q8 | C8⋊D5 | S3×C8 | C5⋊2C8 | C40 | C4×D5 | C3⋊C8 | C24 | C4×S3 | Dic5 | D10 | C15 | Dic3 | D6 | C3 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 16 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C40.55D6 ►in GL4(𝔽241) generated by
| 70 | 70 | 0 | 0 |
| 171 | 115 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 177 |
| 1 | 51 | 0 | 0 |
| 0 | 240 | 0 | 0 |
| 0 | 0 | 239 | 49 |
| 0 | 0 | 177 | 1 |
| 76 | 69 | 0 | 0 |
| 49 | 165 | 0 | 0 |
| 0 | 0 | 35 | 115 |
| 0 | 0 | 113 | 206 |
G:=sub<GL(4,GF(241))| [70,171,0,0,70,115,0,0,0,0,177,0,0,0,0,177],[1,0,0,0,51,240,0,0,0,0,239,177,0,0,49,1],[76,49,0,0,69,165,0,0,0,0,35,113,0,0,115,206] >;
C40.55D6 in GAP, Magma, Sage, TeX
C_{40}._{55}D_6 % in TeX
G:=Group("C40.55D6"); // GroupNames label
G:=SmallGroup(480,343);
// by ID
G=gap.SmallGroup(480,343);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,422,58,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^40=b^6=1,c^2=a^20,b*a*b^-1=c*a*c^-1=a^29,c*b*c^-1=a^20*b^-1>;
// generators/relations