metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊3C8, C4.19D60, C12.37D20, C20.37D12, C60.209D4, C30.27M4(2), (C2×C40)⋊3S3, C5⋊4(D6⋊C8), (C2×C24)⋊3D5, (C2×C8)⋊1D15, (C2×C120)⋊5C2, C2.5(C8×D15), C6.10(C8×D5), C10.19(S3×C8), C30.47(C2×C8), (C2×C4).94D30, C3⋊2(D10⋊1C8), C15⋊10(C22⋊C8), (C2×C20).408D6, C6.6(C8⋊D5), C2.3(C40⋊S3), C10.32(D6⋊C4), (C2×C12).412D10, C4.27(C15⋊7D4), (C22×D15).8C4, C22.11(C4×D15), C10.11(C8⋊S3), C2.1(D30⋊3C4), C20.106(C3⋊D4), C12.106(C5⋊D4), C30.74(C22⋊C4), (C2×C60).494C22, (C2×Dic15).14C4, C6.17(D10⋊C4), (C2×C4×D15).6C2, (C2×C15⋊3C8)⋊9C2, (C2×C6).29(C4×D5), (C2×C10).54(C4×S3), (C2×C30).136(C2×C4), SmallGroup(480,180)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊3C8
G = < a,b,c | a30=b2=c8=1, bab=a-1, ac=ca, cbc-1=a15b >
Subgroups: 596 in 100 conjugacy classes, 43 normal (41 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C8, C2×C4, C2×C4, C23, D5, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, C2×C8, C22×C4, Dic5, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, C2×Dic3, C2×C12, C22×S3, D15, C30, C22⋊C8, C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, C2×C24, S3×C2×C4, Dic15, C60, D30, D30, C2×C30, C2×C5⋊2C8, C2×C40, C2×C4×D5, D6⋊C8, C15⋊3C8, C120, C4×D15, C2×Dic15, C2×C60, C22×D15, D10⋊1C8, C2×C15⋊3C8, C2×C120, C2×C4×D15, D30⋊3C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, D5, D6, C22⋊C4, C2×C8, M4(2), D10, C4×S3, D12, C3⋊D4, D15, C22⋊C8, C4×D5, D20, C5⋊D4, S3×C8, C8⋊S3, D6⋊C4, D30, C8×D5, C8⋊D5, D10⋊C4, D6⋊C8, C4×D15, D60, C15⋊7D4, D10⋊1C8, C8×D15, C40⋊S3, D30⋊3C4, D30⋊3C8
►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 57)(2 56)(3 55)(4 54)(5 53)(6 52)(7 51)(8 50)(9 49)(10 48)(11 47)(12 46)(13 45)(14 44)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 60)(29 59)(30 58)(61 98)(62 97)(63 96)(64 95)(65 94)(66 93)(67 92)(68 91)(69 120)(70 119)(71 118)(72 117)(73 116)(74 115)(75 114)(76 113)(77 112)(78 111)(79 110)(80 109)(81 108)(82 107)(83 106)(84 105)(85 104)(86 103)(87 102)(88 101)(89 100)(90 99)(121 236)(122 235)(123 234)(124 233)(125 232)(126 231)(127 230)(128 229)(129 228)(130 227)(131 226)(132 225)(133 224)(134 223)(135 222)(136 221)(137 220)(138 219)(139 218)(140 217)(141 216)(142 215)(143 214)(144 213)(145 212)(146 211)(147 240)(148 239)(149 238)(150 237)(151 202)(152 201)(153 200)(154 199)(155 198)(156 197)(157 196)(158 195)(159 194)(160 193)(161 192)(162 191)(163 190)(164 189)(165 188)(166 187)(167 186)(168 185)(169 184)(170 183)(171 182)(172 181)(173 210)(174 209)(175 208)(176 207)(177 206)(178 205)(179 204)(180 203)
(1 196 114 133 58 173 76 240)(2 197 115 134 59 174 77 211)(3 198 116 135 60 175 78 212)(4 199 117 136 31 176 79 213)(5 200 118 137 32 177 80 214)(6 201 119 138 33 178 81 215)(7 202 120 139 34 179 82 216)(8 203 91 140 35 180 83 217)(9 204 92 141 36 151 84 218)(10 205 93 142 37 152 85 219)(11 206 94 143 38 153 86 220)(12 207 95 144 39 154 87 221)(13 208 96 145 40 155 88 222)(14 209 97 146 41 156 89 223)(15 210 98 147 42 157 90 224)(16 181 99 148 43 158 61 225)(17 182 100 149 44 159 62 226)(18 183 101 150 45 160 63 227)(19 184 102 121 46 161 64 228)(20 185 103 122 47 162 65 229)(21 186 104 123 48 163 66 230)(22 187 105 124 49 164 67 231)(23 188 106 125 50 165 68 232)(24 189 107 126 51 166 69 233)(25 190 108 127 52 167 70 234)(26 191 109 128 53 168 71 235)(27 192 110 129 54 169 72 236)(28 193 111 130 55 170 73 237)(29 194 112 131 56 171 74 238)(30 195 113 132 57 172 75 239)
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), (1,57)(2,56)(3,55)(4,54)(5,53)(6,52)(7,51)(8,50)(9,49)(10,48)(11,47)(12,46)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,60)(29,59)(30,58)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,120)(70,119)(71,118)(72,117)(73,116)(74,115)(75,114)(76,113)(77,112)(78,111)(79,110)(80,109)(81,108)(82,107)(83,106)(84,105)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(121,236)(122,235)(123,234)(124,233)(125,232)(126,231)(127,230)(128,229)(129,228)(130,227)(131,226)(132,225)(133,224)(134,223)(135,222)(136,221)(137,220)(138,219)(139,218)(140,217)(141,216)(142,215)(143,214)(144,213)(145,212)(146,211)(147,240)(148,239)(149,238)(150,237)(151,202)(152,201)(153,200)(154,199)(155,198)(156,197)(157,196)(158,195)(159,194)(160,193)(161,192)(162,191)(163,190)(164,189)(165,188)(166,187)(167,186)(168,185)(169,184)(170,183)(171,182)(172,181)(173,210)(174,209)(175,208)(176,207)(177,206)(178,205)(179,204)(180,203), (1,196,114,133,58,173,76,240)(2,197,115,134,59,174,77,211)(3,198,116,135,60,175,78,212)(4,199,117,136,31,176,79,213)(5,200,118,137,32,177,80,214)(6,201,119,138,33,178,81,215)(7,202,120,139,34,179,82,216)(8,203,91,140,35,180,83,217)(9,204,92,141,36,151,84,218)(10,205,93,142,37,152,85,219)(11,206,94,143,38,153,86,220)(12,207,95,144,39,154,87,221)(13,208,96,145,40,155,88,222)(14,209,97,146,41,156,89,223)(15,210,98,147,42,157,90,224)(16,181,99,148,43,158,61,225)(17,182,100,149,44,159,62,226)(18,183,101,150,45,160,63,227)(19,184,102,121,46,161,64,228)(20,185,103,122,47,162,65,229)(21,186,104,123,48,163,66,230)(22,187,105,124,49,164,67,231)(23,188,106,125,50,165,68,232)(24,189,107,126,51,166,69,233)(25,190,108,127,52,167,70,234)(26,191,109,128,53,168,71,235)(27,192,110,129,54,169,72,236)(28,193,111,130,55,170,73,237)(29,194,112,131,56,171,74,238)(30,195,113,132,57,172,75,239)>;
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), (1,57)(2,56)(3,55)(4,54)(5,53)(6,52)(7,51)(8,50)(9,49)(10,48)(11,47)(12,46)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,60)(29,59)(30,58)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,120)(70,119)(71,118)(72,117)(73,116)(74,115)(75,114)(76,113)(77,112)(78,111)(79,110)(80,109)(81,108)(82,107)(83,106)(84,105)(85,104)(86,103)(87,102)(88,101)(89,100)(90,99)(121,236)(122,235)(123,234)(124,233)(125,232)(126,231)(127,230)(128,229)(129,228)(130,227)(131,226)(132,225)(133,224)(134,223)(135,222)(136,221)(137,220)(138,219)(139,218)(140,217)(141,216)(142,215)(143,214)(144,213)(145,212)(146,211)(147,240)(148,239)(149,238)(150,237)(151,202)(152,201)(153,200)(154,199)(155,198)(156,197)(157,196)(158,195)(159,194)(160,193)(161,192)(162,191)(163,190)(164,189)(165,188)(166,187)(167,186)(168,185)(169,184)(170,183)(171,182)(172,181)(173,210)(174,209)(175,208)(176,207)(177,206)(178,205)(179,204)(180,203), (1,196,114,133,58,173,76,240)(2,197,115,134,59,174,77,211)(3,198,116,135,60,175,78,212)(4,199,117,136,31,176,79,213)(5,200,118,137,32,177,80,214)(6,201,119,138,33,178,81,215)(7,202,120,139,34,179,82,216)(8,203,91,140,35,180,83,217)(9,204,92,141,36,151,84,218)(10,205,93,142,37,152,85,219)(11,206,94,143,38,153,86,220)(12,207,95,144,39,154,87,221)(13,208,96,145,40,155,88,222)(14,209,97,146,41,156,89,223)(15,210,98,147,42,157,90,224)(16,181,99,148,43,158,61,225)(17,182,100,149,44,159,62,226)(18,183,101,150,45,160,63,227)(19,184,102,121,46,161,64,228)(20,185,103,122,47,162,65,229)(21,186,104,123,48,163,66,230)(22,187,105,124,49,164,67,231)(23,188,106,125,50,165,68,232)(24,189,107,126,51,166,69,233)(25,190,108,127,52,167,70,234)(26,191,109,128,53,168,71,235)(27,192,110,129,54,169,72,236)(28,193,111,130,55,170,73,237)(29,194,112,131,56,171,74,238)(30,195,113,132,57,172,75,239) );
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)], [(1,57),(2,56),(3,55),(4,54),(5,53),(6,52),(7,51),(8,50),(9,49),(10,48),(11,47),(12,46),(13,45),(14,44),(15,43),(16,42),(17,41),(18,40),(19,39),(20,38),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,60),(29,59),(30,58),(61,98),(62,97),(63,96),(64,95),(65,94),(66,93),(67,92),(68,91),(69,120),(70,119),(71,118),(72,117),(73,116),(74,115),(75,114),(76,113),(77,112),(78,111),(79,110),(80,109),(81,108),(82,107),(83,106),(84,105),(85,104),(86,103),(87,102),(88,101),(89,100),(90,99),(121,236),(122,235),(123,234),(124,233),(125,232),(126,231),(127,230),(128,229),(129,228),(130,227),(131,226),(132,225),(133,224),(134,223),(135,222),(136,221),(137,220),(138,219),(139,218),(140,217),(141,216),(142,215),(143,214),(144,213),(145,212),(146,211),(147,240),(148,239),(149,238),(150,237),(151,202),(152,201),(153,200),(154,199),(155,198),(156,197),(157,196),(158,195),(159,194),(160,193),(161,192),(162,191),(163,190),(164,189),(165,188),(166,187),(167,186),(168,185),(169,184),(170,183),(171,182),(172,181),(173,210),(174,209),(175,208),(176,207),(177,206),(178,205),(179,204),(180,203)], [(1,196,114,133,58,173,76,240),(2,197,115,134,59,174,77,211),(3,198,116,135,60,175,78,212),(4,199,117,136,31,176,79,213),(5,200,118,137,32,177,80,214),(6,201,119,138,33,178,81,215),(7,202,120,139,34,179,82,216),(8,203,91,140,35,180,83,217),(9,204,92,141,36,151,84,218),(10,205,93,142,37,152,85,219),(11,206,94,143,38,153,86,220),(12,207,95,144,39,154,87,221),(13,208,96,145,40,155,88,222),(14,209,97,146,41,156,89,223),(15,210,98,147,42,157,90,224),(16,181,99,148,43,158,61,225),(17,182,100,149,44,159,62,226),(18,183,101,150,45,160,63,227),(19,184,102,121,46,161,64,228),(20,185,103,122,47,162,65,229),(21,186,104,123,48,163,66,230),(22,187,105,124,49,164,67,231),(23,188,106,125,50,165,68,232),(24,189,107,126,51,166,69,233),(25,190,108,127,52,167,70,234),(26,191,109,128,53,168,71,235),(27,192,110,129,54,169,72,236),(28,193,111,130,55,170,73,237),(29,194,112,131,56,171,74,238),(30,195,113,132,57,172,75,239)]])
132 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 15A | 15B | 15C | 15D | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30L | 40A | ··· | 40P | 60A | ··· | 60P | 120A | ··· | 120AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
132 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | D5 | D6 | M4(2) | D10 | D12 | C3⋊D4 | C4×S3 | D15 | D20 | C5⋊D4 | C4×D5 | S3×C8 | C8⋊S3 | D30 | C8×D5 | C8⋊D5 | D60 | C15⋊7D4 | C4×D15 | C8×D15 | C40⋊S3 |
| kernel | D30⋊3C8 | C2×C15⋊3C8 | C2×C120 | C2×C4×D15 | C2×Dic15 | C22×D15 | D30 | C2×C40 | C60 | C2×C24 | C2×C20 | C30 | C2×C12 | C20 | C20 | C2×C10 | C2×C8 | C12 | C12 | C2×C6 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | 16 | 16 |
Matrix representation of D30⋊3C8 ►in GL3(𝔽241) generated by
| 1 | 0 | 0 |
| 0 | 173 | 63 |
| 0 | 178 | 30 |
| 1 | 0 | 0 |
| 0 | 173 | 63 |
| 0 | 225 | 68 |
| 211 | 0 | 0 |
| 0 | 154 | 43 |
| 0 | 198 | 87 |
G:=sub<GL(3,GF(241))| [1,0,0,0,173,178,0,63,30],[1,0,0,0,173,225,0,63,68],[211,0,0,0,154,198,0,43,87] >;
D30⋊3C8 in GAP, Magma, Sage, TeX
D_{30}\rtimes_3C_8 % in TeX
G:=Group("D30:3C8"); // GroupNames label
G:=SmallGroup(480,180);
// by ID
G=gap.SmallGroup(480,180);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,36,100,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^30=b^2=c^8=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^15*b>;
// generators/relations