metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊1C8, C30.3M4(2), Dic5.23D12, C5⋊1(D6⋊C8), C10.3(S3×C8), C30.7(C2×C8), C15⋊2(C22⋊C8), C6.4(D5⋊C8), C2.3(D6⋊F5), C3⋊1(D10⋊C8), C6.2(C4.F5), C2.4(D15⋊C8), (C2×Dic3).3F5, C10.2(C8⋊S3), C10.11(D6⋊C4), C22.14(S3×F5), (C3×Dic5).31D4, (C10×Dic3).4C4, (C22×D15).3C4, C6.11(C22⋊F5), C30.11(C22⋊C4), (C2×Dic5).144D6, C2.2(Dic3.F5), Dic5.29(C3⋊D4), (C6×Dic5).137C22, (C6×C5⋊C8)⋊2C2, (C2×C5⋊C8)⋊2S3, (C2×C15⋊C8)⋊2C2, (C2×C10).7(C4×S3), (C2×C30).5(C2×C4), (C2×C6).15(C2×F5), (C2×D30.C2).10C2, SmallGroup(480,247)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊C8
G = < a,b,c | a30=b2=c8=1, bab=a-1, cac-1=a13, cbc-1=a27b >
Subgroups: 596 in 100 conjugacy classes, 36 normal (34 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C8, C2×C4, C23, D5, C10, Dic3, C12, D6, C2×C6, C15, 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⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C2×C3⋊C8, C2×C24, S3×C2×C4, C5×Dic3, C3×Dic5, D30, D30, C2×C30, C2×C5⋊C8, C2×C5⋊C8, C2×C4×D5, D6⋊C8, C3×C5⋊C8, C15⋊C8, D30.C2, C6×Dic5, C10×Dic3, C22×D15, D10⋊C8, C6×C5⋊C8, C2×C15⋊C8, C2×D30.C2, D30⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, D6, C22⋊C4, C2×C8, M4(2), F5, C4×S3, D12, C3⋊D4, C22⋊C8, C2×F5, S3×C8, C8⋊S3, D6⋊C4, D5⋊C8, C4.F5, C22⋊F5, D6⋊C8, S3×F5, D10⋊C8, D6⋊F5, D15⋊C8, Dic3.F5, D30⋊C8
►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)
(1 183)(2 182)(3 181)(4 210)(5 209)(6 208)(7 207)(8 206)(9 205)(10 204)(11 203)(12 202)(13 201)(14 200)(15 199)(16 198)(17 197)(18 196)(19 195)(20 194)(21 193)(22 192)(23 191)(24 190)(25 189)(26 188)(27 187)(28 186)(29 185)(30 184)(31 74)(32 73)(33 72)(34 71)(35 70)(36 69)(37 68)(38 67)(39 66)(40 65)(41 64)(42 63)(43 62)(44 61)(45 90)(46 89)(47 88)(48 87)(49 86)(50 85)(51 84)(52 83)(53 82)(54 81)(55 80)(56 79)(57 78)(58 77)(59 76)(60 75)(91 212)(92 211)(93 240)(94 239)(95 238)(96 237)(97 236)(98 235)(99 234)(100 233)(101 232)(102 231)(103 230)(104 229)(105 228)(106 227)(107 226)(108 225)(109 224)(110 223)(111 222)(112 221)(113 220)(114 219)(115 218)(116 217)(117 216)(118 215)(119 214)(120 213)(121 153)(122 152)(123 151)(124 180)(125 179)(126 178)(127 177)(128 176)(129 175)(130 174)(131 173)(132 172)(133 171)(134 170)(135 169)(136 168)(137 167)(138 166)(139 165)(140 164)(141 163)(142 162)(143 161)(144 160)(145 159)(146 158)(147 157)(148 156)(149 155)(150 154)
(1 146 234 69 184 174 100 52)(2 123 223 82 185 151 119 35)(3 130 212 65 186 158 108 48)(4 137 231 78 187 165 97 31)(5 144 220 61 188 172 116 44)(6 121 239 74 189 179 105 57)(7 128 228 87 190 156 94 40)(8 135 217 70 191 163 113 53)(9 142 236 83 192 170 102 36)(10 149 225 66 193 177 91 49)(11 126 214 79 194 154 110 32)(12 133 233 62 195 161 99 45)(13 140 222 75 196 168 118 58)(14 147 211 88 197 175 107 41)(15 124 230 71 198 152 96 54)(16 131 219 84 199 159 115 37)(17 138 238 67 200 166 104 50)(18 145 227 80 201 173 93 33)(19 122 216 63 202 180 112 46)(20 129 235 76 203 157 101 59)(21 136 224 89 204 164 120 42)(22 143 213 72 205 171 109 55)(23 150 232 85 206 178 98 38)(24 127 221 68 207 155 117 51)(25 134 240 81 208 162 106 34)(26 141 229 64 209 169 95 47)(27 148 218 77 210 176 114 60)(28 125 237 90 181 153 103 43)(29 132 226 73 182 160 92 56)(30 139 215 86 183 167 111 39)
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), (1,183)(2,182)(3,181)(4,210)(5,209)(6,208)(7,207)(8,206)(9,205)(10,204)(11,203)(12,202)(13,201)(14,200)(15,199)(16,198)(17,197)(18,196)(19,195)(20,194)(21,193)(22,192)(23,191)(24,190)(25,189)(26,188)(27,187)(28,186)(29,185)(30,184)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(91,212)(92,211)(93,240)(94,239)(95,238)(96,237)(97,236)(98,235)(99,234)(100,233)(101,232)(102,231)(103,230)(104,229)(105,228)(106,227)(107,226)(108,225)(109,224)(110,223)(111,222)(112,221)(113,220)(114,219)(115,218)(116,217)(117,216)(118,215)(119,214)(120,213)(121,153)(122,152)(123,151)(124,180)(125,179)(126,178)(127,177)(128,176)(129,175)(130,174)(131,173)(132,172)(133,171)(134,170)(135,169)(136,168)(137,167)(138,166)(139,165)(140,164)(141,163)(142,162)(143,161)(144,160)(145,159)(146,158)(147,157)(148,156)(149,155)(150,154), (1,146,234,69,184,174,100,52)(2,123,223,82,185,151,119,35)(3,130,212,65,186,158,108,48)(4,137,231,78,187,165,97,31)(5,144,220,61,188,172,116,44)(6,121,239,74,189,179,105,57)(7,128,228,87,190,156,94,40)(8,135,217,70,191,163,113,53)(9,142,236,83,192,170,102,36)(10,149,225,66,193,177,91,49)(11,126,214,79,194,154,110,32)(12,133,233,62,195,161,99,45)(13,140,222,75,196,168,118,58)(14,147,211,88,197,175,107,41)(15,124,230,71,198,152,96,54)(16,131,219,84,199,159,115,37)(17,138,238,67,200,166,104,50)(18,145,227,80,201,173,93,33)(19,122,216,63,202,180,112,46)(20,129,235,76,203,157,101,59)(21,136,224,89,204,164,120,42)(22,143,213,72,205,171,109,55)(23,150,232,85,206,178,98,38)(24,127,221,68,207,155,117,51)(25,134,240,81,208,162,106,34)(26,141,229,64,209,169,95,47)(27,148,218,77,210,176,114,60)(28,125,237,90,181,153,103,43)(29,132,226,73,182,160,92,56)(30,139,215,86,183,167,111,39)>;
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), (1,183)(2,182)(3,181)(4,210)(5,209)(6,208)(7,207)(8,206)(9,205)(10,204)(11,203)(12,202)(13,201)(14,200)(15,199)(16,198)(17,197)(18,196)(19,195)(20,194)(21,193)(22,192)(23,191)(24,190)(25,189)(26,188)(27,187)(28,186)(29,185)(30,184)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,90)(46,89)(47,88)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(91,212)(92,211)(93,240)(94,239)(95,238)(96,237)(97,236)(98,235)(99,234)(100,233)(101,232)(102,231)(103,230)(104,229)(105,228)(106,227)(107,226)(108,225)(109,224)(110,223)(111,222)(112,221)(113,220)(114,219)(115,218)(116,217)(117,216)(118,215)(119,214)(120,213)(121,153)(122,152)(123,151)(124,180)(125,179)(126,178)(127,177)(128,176)(129,175)(130,174)(131,173)(132,172)(133,171)(134,170)(135,169)(136,168)(137,167)(138,166)(139,165)(140,164)(141,163)(142,162)(143,161)(144,160)(145,159)(146,158)(147,157)(148,156)(149,155)(150,154), (1,146,234,69,184,174,100,52)(2,123,223,82,185,151,119,35)(3,130,212,65,186,158,108,48)(4,137,231,78,187,165,97,31)(5,144,220,61,188,172,116,44)(6,121,239,74,189,179,105,57)(7,128,228,87,190,156,94,40)(8,135,217,70,191,163,113,53)(9,142,236,83,192,170,102,36)(10,149,225,66,193,177,91,49)(11,126,214,79,194,154,110,32)(12,133,233,62,195,161,99,45)(13,140,222,75,196,168,118,58)(14,147,211,88,197,175,107,41)(15,124,230,71,198,152,96,54)(16,131,219,84,199,159,115,37)(17,138,238,67,200,166,104,50)(18,145,227,80,201,173,93,33)(19,122,216,63,202,180,112,46)(20,129,235,76,203,157,101,59)(21,136,224,89,204,164,120,42)(22,143,213,72,205,171,109,55)(23,150,232,85,206,178,98,38)(24,127,221,68,207,155,117,51)(25,134,240,81,208,162,106,34)(26,141,229,64,209,169,95,47)(27,148,218,77,210,176,114,60)(28,125,237,90,181,153,103,43)(29,132,226,73,182,160,92,56)(30,139,215,86,183,167,111,39) );
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)], [(1,183),(2,182),(3,181),(4,210),(5,209),(6,208),(7,207),(8,206),(9,205),(10,204),(11,203),(12,202),(13,201),(14,200),(15,199),(16,198),(17,197),(18,196),(19,195),(20,194),(21,193),(22,192),(23,191),(24,190),(25,189),(26,188),(27,187),(28,186),(29,185),(30,184),(31,74),(32,73),(33,72),(34,71),(35,70),(36,69),(37,68),(38,67),(39,66),(40,65),(41,64),(42,63),(43,62),(44,61),(45,90),(46,89),(47,88),(48,87),(49,86),(50,85),(51,84),(52,83),(53,82),(54,81),(55,80),(56,79),(57,78),(58,77),(59,76),(60,75),(91,212),(92,211),(93,240),(94,239),(95,238),(96,237),(97,236),(98,235),(99,234),(100,233),(101,232),(102,231),(103,230),(104,229),(105,228),(106,227),(107,226),(108,225),(109,224),(110,223),(111,222),(112,221),(113,220),(114,219),(115,218),(116,217),(117,216),(118,215),(119,214),(120,213),(121,153),(122,152),(123,151),(124,180),(125,179),(126,178),(127,177),(128,176),(129,175),(130,174),(131,173),(132,172),(133,171),(134,170),(135,169),(136,168),(137,167),(138,166),(139,165),(140,164),(141,163),(142,162),(143,161),(144,160),(145,159),(146,158),(147,157),(148,156),(149,155),(150,154)], [(1,146,234,69,184,174,100,52),(2,123,223,82,185,151,119,35),(3,130,212,65,186,158,108,48),(4,137,231,78,187,165,97,31),(5,144,220,61,188,172,116,44),(6,121,239,74,189,179,105,57),(7,128,228,87,190,156,94,40),(8,135,217,70,191,163,113,53),(9,142,236,83,192,170,102,36),(10,149,225,66,193,177,91,49),(11,126,214,79,194,154,110,32),(12,133,233,62,195,161,99,45),(13,140,222,75,196,168,118,58),(14,147,211,88,197,175,107,41),(15,124,230,71,198,152,96,54),(16,131,219,84,199,159,115,37),(17,138,238,67,200,166,104,50),(18,145,227,80,201,173,93,33),(19,122,216,63,202,180,112,46),(20,129,235,76,203,157,101,59),(21,136,224,89,204,164,120,42),(22,143,213,72,205,171,109,55),(23,150,232,85,206,178,98,38),(24,127,221,68,207,155,117,51),(25,134,240,81,208,162,106,34),(26,141,229,64,209,169,95,47),(27,148,218,77,210,176,114,60),(28,125,237,90,181,153,103,43),(29,132,226,73,182,160,92,56),(30,139,215,86,183,167,111,39)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 12A | 12B | 12C | 12D | 15 | 20A | 20B | 20C | 20D | 24A | ··· | 24H | 30A | 30B | 30C |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | 30 | 30 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 5 | 5 | 5 | 5 | 6 | 6 | 4 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 8 | 12 | 12 | 12 | 12 | 10 | ··· | 10 | 8 | 8 | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | D6 | M4(2) | D12 | C3⋊D4 | C4×S3 | S3×C8 | C8⋊S3 | F5 | C2×F5 | D5⋊C8 | C4.F5 | C22⋊F5 | S3×F5 | D6⋊F5 | D15⋊C8 | Dic3.F5 |
| kernel | D30⋊C8 | C6×C5⋊C8 | C2×C15⋊C8 | C2×D30.C2 | C10×Dic3 | C22×D15 | D30 | C2×C5⋊C8 | C3×Dic5 | C2×Dic5 | C30 | Dic5 | Dic5 | C2×C10 | C10 | C10 | C2×Dic3 | C2×C6 | C6 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 |
Matrix representation of D30⋊C8 ►in GL8(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 240 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 64 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 142 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 198 | 99 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 68 | 173 | 238 | 0 |
| 0 | 0 | 0 | 0 | 65 | 173 | 0 | 68 |
| 0 | 0 | 0 | 0 | 68 | 0 | 173 | 65 |
| 0 | 0 | 0 | 0 | 0 | 238 | 173 | 68 |
G:=sub<GL(8,GF(241))| [240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,240,1,0,0,0,0,0,0,0,0,240,240,240,240,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,240,240,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,142,198,0,0,0,0,0,0,43,99,0,0,0,0,0,0,0,0,68,65,68,0,0,0,0,0,173,173,0,238,0,0,0,0,238,0,173,173,0,0,0,0,0,68,65,68] >;
D30⋊C8 in GAP, Magma, Sage, TeX
D_{30}\rtimes C_8 % in TeX
G:=Group("D30:C8"); // GroupNames label
G:=SmallGroup(480,247);
// by ID
G=gap.SmallGroup(480,247);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,100,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c|a^30=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^27*b>;
// generators/relations