metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.2C8, C15⋊2M5(2), Dic15.2C8, C5⋊C16⋊2S3, C3⋊C8.3F5, C10.2(S3×C8), C30.4(C2×C8), C5⋊1(D6.C8), C15⋊C16⋊4C2, C4.29(S3×F5), C20.29(C4×S3), C60.29(C2×C4), C6.2(D5⋊C8), (C4×D15).4C4, C5⋊2C8.30D6, C12.36(C2×F5), C3⋊1(C8.F5), C2.3(D15⋊C8), D15⋊2C8.3C2, (C3×C5⋊C16)⋊4C2, (C5×C3⋊C8).4C4, (C3×C5⋊2C8).30C22, SmallGroup(480,242)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30.C8
G = < a,b,c | a30=b2=1, c8=a15, bab=a-1, cac-1=a13, cbc-1=a27b >
30C2
15C4
15C22
10S3
6D5
3C8
5C8
15C2×C4
5D6
5Dic3
3D10
3Dic5
2D15
5C16
15C2×C8
15C16
5C24
3C40
15M5(2)
5C48
Smallest permutation representation of 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 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(31 60)(32 59)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(61 88)(62 87)(63 86)(64 85)(65 84)(66 83)(67 82)(68 81)(69 80)(70 79)(71 78)(72 77)(73 76)(74 75)(89 90)(91 116)(92 115)(93 114)(94 113)(95 112)(96 111)(97 110)(98 109)(99 108)(100 107)(101 106)(102 105)(103 104)(117 120)(118 119)(121 123)(124 150)(125 149)(126 148)(127 147)(128 146)(129 145)(130 144)(131 143)(132 142)(133 141)(134 140)(135 139)(136 138)(151 157)(152 156)(153 155)(158 180)(159 179)(160 178)(161 177)(162 176)(163 175)(164 174)(165 173)(166 172)(167 171)(168 170)(181 189)(182 188)(183 187)(184 186)(190 210)(191 209)(192 208)(193 207)(194 206)(195 205)(196 204)(197 203)(198 202)(199 201)(211 219)(212 218)(213 217)(214 216)(220 240)(221 239)(222 238)(223 237)(224 236)(225 235)(226 234)(227 233)(228 232)(229 231)
(1 238 104 177 46 208 75 145 16 223 119 162 31 193 90 130)(2 215 93 160 47 185 64 128 17 230 108 175 32 200 79 143)(3 222 112 173 48 192 83 141 18 237 97 158 33 207 68 126)(4 229 101 156 49 199 72 124 19 214 116 171 34 184 87 139)(5 236 120 169 50 206 61 137 20 221 105 154 35 191 76 122)(6 213 109 152 51 183 80 150 21 228 94 167 36 198 65 135)(7 220 98 165 52 190 69 133 22 235 113 180 37 205 84 148)(8 227 117 178 53 197 88 146 23 212 102 163 38 182 73 131)(9 234 106 161 54 204 77 129 24 219 91 176 39 189 62 144)(10 211 95 174 55 181 66 142 25 226 110 159 40 196 81 127)(11 218 114 157 56 188 85 125 26 233 99 172 41 203 70 140)(12 225 103 170 57 195 74 138 27 240 118 155 42 210 89 123)(13 232 92 153 58 202 63 121 28 217 107 168 43 187 78 136)(14 239 111 166 59 209 82 134 29 224 96 151 44 194 67 149)(15 216 100 179 60 186 71 147 30 231 115 164 45 201 86 132)
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(89,90)(91,116)(92,115)(93,114)(94,113)(95,112)(96,111)(97,110)(98,109)(99,108)(100,107)(101,106)(102,105)(103,104)(117,120)(118,119)(121,123)(124,150)(125,149)(126,148)(127,147)(128,146)(129,145)(130,144)(131,143)(132,142)(133,141)(134,140)(135,139)(136,138)(151,157)(152,156)(153,155)(158,180)(159,179)(160,178)(161,177)(162,176)(163,175)(164,174)(165,173)(166,172)(167,171)(168,170)(181,189)(182,188)(183,187)(184,186)(190,210)(191,209)(192,208)(193,207)(194,206)(195,205)(196,204)(197,203)(198,202)(199,201)(211,219)(212,218)(213,217)(214,216)(220,240)(221,239)(222,238)(223,237)(224,236)(225,235)(226,234)(227,233)(228,232)(229,231), (1,238,104,177,46,208,75,145,16,223,119,162,31,193,90,130)(2,215,93,160,47,185,64,128,17,230,108,175,32,200,79,143)(3,222,112,173,48,192,83,141,18,237,97,158,33,207,68,126)(4,229,101,156,49,199,72,124,19,214,116,171,34,184,87,139)(5,236,120,169,50,206,61,137,20,221,105,154,35,191,76,122)(6,213,109,152,51,183,80,150,21,228,94,167,36,198,65,135)(7,220,98,165,52,190,69,133,22,235,113,180,37,205,84,148)(8,227,117,178,53,197,88,146,23,212,102,163,38,182,73,131)(9,234,106,161,54,204,77,129,24,219,91,176,39,189,62,144)(10,211,95,174,55,181,66,142,25,226,110,159,40,196,81,127)(11,218,114,157,56,188,85,125,26,233,99,172,41,203,70,140)(12,225,103,170,57,195,74,138,27,240,118,155,42,210,89,123)(13,232,92,153,58,202,63,121,28,217,107,168,43,187,78,136)(14,239,111,166,59,209,82,134,29,224,96,151,44,194,67,149)(15,216,100,179,60,186,71,147,30,231,115,164,45,201,86,132)>;
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), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(31,60)(32,59)(33,58)(34,57)(35,56)(36,55)(37,54)(38,53)(39,52)(40,51)(41,50)(42,49)(43,48)(44,47)(45,46)(61,88)(62,87)(63,86)(64,85)(65,84)(66,83)(67,82)(68,81)(69,80)(70,79)(71,78)(72,77)(73,76)(74,75)(89,90)(91,116)(92,115)(93,114)(94,113)(95,112)(96,111)(97,110)(98,109)(99,108)(100,107)(101,106)(102,105)(103,104)(117,120)(118,119)(121,123)(124,150)(125,149)(126,148)(127,147)(128,146)(129,145)(130,144)(131,143)(132,142)(133,141)(134,140)(135,139)(136,138)(151,157)(152,156)(153,155)(158,180)(159,179)(160,178)(161,177)(162,176)(163,175)(164,174)(165,173)(166,172)(167,171)(168,170)(181,189)(182,188)(183,187)(184,186)(190,210)(191,209)(192,208)(193,207)(194,206)(195,205)(196,204)(197,203)(198,202)(199,201)(211,219)(212,218)(213,217)(214,216)(220,240)(221,239)(222,238)(223,237)(224,236)(225,235)(226,234)(227,233)(228,232)(229,231), (1,238,104,177,46,208,75,145,16,223,119,162,31,193,90,130)(2,215,93,160,47,185,64,128,17,230,108,175,32,200,79,143)(3,222,112,173,48,192,83,141,18,237,97,158,33,207,68,126)(4,229,101,156,49,199,72,124,19,214,116,171,34,184,87,139)(5,236,120,169,50,206,61,137,20,221,105,154,35,191,76,122)(6,213,109,152,51,183,80,150,21,228,94,167,36,198,65,135)(7,220,98,165,52,190,69,133,22,235,113,180,37,205,84,148)(8,227,117,178,53,197,88,146,23,212,102,163,38,182,73,131)(9,234,106,161,54,204,77,129,24,219,91,176,39,189,62,144)(10,211,95,174,55,181,66,142,25,226,110,159,40,196,81,127)(11,218,114,157,56,188,85,125,26,233,99,172,41,203,70,140)(12,225,103,170,57,195,74,138,27,240,118,155,42,210,89,123)(13,232,92,153,58,202,63,121,28,217,107,168,43,187,78,136)(14,239,111,166,59,209,82,134,29,224,96,151,44,194,67,149)(15,216,100,179,60,186,71,147,30,231,115,164,45,201,86,132) );
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)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),(31,60),(32,59),(33,58),(34,57),(35,56),(36,55),(37,54),(38,53),(39,52),(40,51),(41,50),(42,49),(43,48),(44,47),(45,46),(61,88),(62,87),(63,86),(64,85),(65,84),(66,83),(67,82),(68,81),(69,80),(70,79),(71,78),(72,77),(73,76),(74,75),(89,90),(91,116),(92,115),(93,114),(94,113),(95,112),(96,111),(97,110),(98,109),(99,108),(100,107),(101,106),(102,105),(103,104),(117,120),(118,119),(121,123),(124,150),(125,149),(126,148),(127,147),(128,146),(129,145),(130,144),(131,143),(132,142),(133,141),(134,140),(135,139),(136,138),(151,157),(152,156),(153,155),(158,180),(159,179),(160,178),(161,177),(162,176),(163,175),(164,174),(165,173),(166,172),(167,171),(168,170),(181,189),(182,188),(183,187),(184,186),(190,210),(191,209),(192,208),(193,207),(194,206),(195,205),(196,204),(197,203),(198,202),(199,201),(211,219),(212,218),(213,217),(214,216),(220,240),(221,239),(222,238),(223,237),(224,236),(225,235),(226,234),(227,233),(228,232),(229,231)], [(1,238,104,177,46,208,75,145,16,223,119,162,31,193,90,130),(2,215,93,160,47,185,64,128,17,230,108,175,32,200,79,143),(3,222,112,173,48,192,83,141,18,237,97,158,33,207,68,126),(4,229,101,156,49,199,72,124,19,214,116,171,34,184,87,139),(5,236,120,169,50,206,61,137,20,221,105,154,35,191,76,122),(6,213,109,152,51,183,80,150,21,228,94,167,36,198,65,135),(7,220,98,165,52,190,69,133,22,235,113,180,37,205,84,148),(8,227,117,178,53,197,88,146,23,212,102,163,38,182,73,131),(9,234,106,161,54,204,77,129,24,219,91,176,39,189,62,144),(10,211,95,174,55,181,66,142,25,226,110,159,40,196,81,127),(11,218,114,157,56,188,85,125,26,233,99,172,41,203,70,140),(12,225,103,170,57,195,74,138,27,240,118,155,42,210,89,123),(13,232,92,153,58,202,63,121,28,217,107,168,43,187,78,136),(14,239,111,166,59,209,82,134,29,224,96,151,44,194,67,149),(15,216,100,179,60,186,71,147,30,231,115,164,45,201,86,132)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5 | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 10 | 12A | 12B | 15 | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 24A | 24B | 24C | 24D | 30 | 40A | 40B | 40C | 40D | 48A | ··· | 48H | 60A | 60B |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 12 | 12 | 15 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 40 | 40 | 40 | 40 | 48 | ··· | 48 | 60 | 60 |
| size | 1 | 1 | 30 | 2 | 1 | 1 | 30 | 4 | 2 | 5 | 5 | 5 | 5 | 6 | 6 | 4 | 2 | 2 | 8 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 4 | 4 | 10 | 10 | 10 | 10 | 8 | 12 | 12 | 12 | 12 | 10 | ··· | 10 | 8 | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | D6 | C4×S3 | M5(2) | S3×C8 | D6.C8 | F5 | C2×F5 | D5⋊C8 | C8.F5 | S3×F5 | D15⋊C8 | D30.C8 |
| kernel | D30.C8 | C3×C5⋊C16 | C15⋊C16 | D15⋊2C8 | C5×C3⋊C8 | C4×D15 | Dic15 | D30 | C5⋊C16 | C5⋊2C8 | C20 | C15 | C10 | C5 | C3⋊C8 | C12 | C6 | C3 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 2 | 4 | 1 | 1 | 2 |
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 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 240 | 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 |
| 240 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 30 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 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 |
| 211 | 239 | 0 | 0 | 0 | 0 | 0 | 0 |
| 205 | 30 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 211 | 30 | 109 | 0 |
| 0 | 0 | 0 | 0 | 79 | 30 | 0 | 211 |
| 0 | 0 | 0 | 0 | 211 | 0 | 30 | 79 |
| 0 | 0 | 0 | 0 | 0 | 109 | 30 | 211 |
G:=sub<GL(8,GF(241))| [240,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,240,0,0,0,0,0,0,1,240,0,0,0,0,0,0,0,0,240,240,240,240,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[240,30,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,240,240,240,240,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[211,205,0,0,0,0,0,0,239,30,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,211,79,211,0,0,0,0,0,30,30,0,109,0,0,0,0,109,0,30,30,0,0,0,0,0,211,79,211] >;
D30.C8 in GAP, Magma, Sage, TeX
D_{30}.C_8 % in TeX
G:=Group("D30.C8"); // GroupNames label
G:=SmallGroup(480,242);
// by ID
G=gap.SmallGroup(480,242);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,64,58,80,1356,9414,4724]);
// Polycyclic
G:=Group<a,b,c|a^30=b^2=1,c^8=a^15,b*a*b=a^-1,c*a*c^-1=a^13,c*b*c^-1=a^27*b>;
// generators/relations
Export