metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.5C8, C40.53D6, C15⋊7M5(2), C24.60D10, Dic15.5C8, C120.56C22, C3⋊C16⋊5D5, C6.2(C8×D5), C5⋊2C16⋊5S3, C5⋊3(D6.C8), C8.39(S3×D5), C3⋊1(C80⋊C2), C15⋊3C8.9C4, C10.11(S3×C8), C20.69(C4×S3), C30.24(C2×C8), (C8×D15).5C2, C12.37(C4×D5), C60.140(C2×C4), (C4×D15).12C4, C2.3(D15⋊2C8), C4.17(D30.C2), (C5×C3⋊C16)⋊7C2, (C3×C5⋊2C16)⋊10C2, SmallGroup(480,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30.5C8
G = < a,b,c | a30=b2=1, c8=a15, bab=a-1, cac-1=a19, cbc-1=a3b >
Smallest permutation representation of D30.5C8
►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 90)(62 89)(63 88)(64 87)(65 86)(66 85)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)(73 78)(74 77)(75 76)(91 120)(92 119)(93 118)(94 117)(95 116)(96 115)(97 114)(98 113)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)(121 145)(122 144)(123 143)(124 142)(125 141)(126 140)(127 139)(128 138)(129 137)(130 136)(131 135)(132 134)(146 150)(147 149)(151 165)(152 164)(153 163)(154 162)(155 161)(156 160)(157 159)(166 180)(167 179)(168 178)(169 177)(170 176)(171 175)(172 174)(181 185)(182 184)(186 210)(187 209)(188 208)(189 207)(190 206)(191 205)(192 204)(193 203)(194 202)(195 201)(196 200)(197 199)(211 215)(212 214)(216 240)(217 239)(218 238)(219 237)(220 236)(221 235)(222 234)(223 233)(224 232)(225 231)(226 230)(227 229)
(1 236 91 166 31 206 61 141 16 221 106 151 46 191 76 126)(2 225 92 155 32 195 62 130 17 240 107 170 47 210 77 145)(3 214 93 174 33 184 63 149 18 229 108 159 48 199 78 134)(4 233 94 163 34 203 64 138 19 218 109 178 49 188 79 123)(5 222 95 152 35 192 65 127 20 237 110 167 50 207 80 142)(6 211 96 171 36 181 66 146 21 226 111 156 51 196 81 131)(7 230 97 160 37 200 67 135 22 215 112 175 52 185 82 150)(8 219 98 179 38 189 68 124 23 234 113 164 53 204 83 139)(9 238 99 168 39 208 69 143 24 223 114 153 54 193 84 128)(10 227 100 157 40 197 70 132 25 212 115 172 55 182 85 147)(11 216 101 176 41 186 71 121 26 231 116 161 56 201 86 136)(12 235 102 165 42 205 72 140 27 220 117 180 57 190 87 125)(13 224 103 154 43 194 73 129 28 239 118 169 58 209 88 144)(14 213 104 173 44 183 74 148 29 228 119 158 59 198 89 133)(15 232 105 162 45 202 75 137 30 217 120 177 60 187 90 122)
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)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(121,145)(122,144)(123,143)(124,142)(125,141)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(146,150)(147,149)(151,165)(152,164)(153,163)(154,162)(155,161)(156,160)(157,159)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(172,174)(181,185)(182,184)(186,210)(187,209)(188,208)(189,207)(190,206)(191,205)(192,204)(193,203)(194,202)(195,201)(196,200)(197,199)(211,215)(212,214)(216,240)(217,239)(218,238)(219,237)(220,236)(221,235)(222,234)(223,233)(224,232)(225,231)(226,230)(227,229), (1,236,91,166,31,206,61,141,16,221,106,151,46,191,76,126)(2,225,92,155,32,195,62,130,17,240,107,170,47,210,77,145)(3,214,93,174,33,184,63,149,18,229,108,159,48,199,78,134)(4,233,94,163,34,203,64,138,19,218,109,178,49,188,79,123)(5,222,95,152,35,192,65,127,20,237,110,167,50,207,80,142)(6,211,96,171,36,181,66,146,21,226,111,156,51,196,81,131)(7,230,97,160,37,200,67,135,22,215,112,175,52,185,82,150)(8,219,98,179,38,189,68,124,23,234,113,164,53,204,83,139)(9,238,99,168,39,208,69,143,24,223,114,153,54,193,84,128)(10,227,100,157,40,197,70,132,25,212,115,172,55,182,85,147)(11,216,101,176,41,186,71,121,26,231,116,161,56,201,86,136)(12,235,102,165,42,205,72,140,27,220,117,180,57,190,87,125)(13,224,103,154,43,194,73,129,28,239,118,169,58,209,88,144)(14,213,104,173,44,183,74,148,29,228,119,158,59,198,89,133)(15,232,105,162,45,202,75,137,30,217,120,177,60,187,90,122)>;
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)(61,90)(62,89)(63,88)(64,87)(65,86)(66,85)(67,84)(68,83)(69,82)(70,81)(71,80)(72,79)(73,78)(74,77)(75,76)(91,120)(92,119)(93,118)(94,117)(95,116)(96,115)(97,114)(98,113)(99,112)(100,111)(101,110)(102,109)(103,108)(104,107)(105,106)(121,145)(122,144)(123,143)(124,142)(125,141)(126,140)(127,139)(128,138)(129,137)(130,136)(131,135)(132,134)(146,150)(147,149)(151,165)(152,164)(153,163)(154,162)(155,161)(156,160)(157,159)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(172,174)(181,185)(182,184)(186,210)(187,209)(188,208)(189,207)(190,206)(191,205)(192,204)(193,203)(194,202)(195,201)(196,200)(197,199)(211,215)(212,214)(216,240)(217,239)(218,238)(219,237)(220,236)(221,235)(222,234)(223,233)(224,232)(225,231)(226,230)(227,229), (1,236,91,166,31,206,61,141,16,221,106,151,46,191,76,126)(2,225,92,155,32,195,62,130,17,240,107,170,47,210,77,145)(3,214,93,174,33,184,63,149,18,229,108,159,48,199,78,134)(4,233,94,163,34,203,64,138,19,218,109,178,49,188,79,123)(5,222,95,152,35,192,65,127,20,237,110,167,50,207,80,142)(6,211,96,171,36,181,66,146,21,226,111,156,51,196,81,131)(7,230,97,160,37,200,67,135,22,215,112,175,52,185,82,150)(8,219,98,179,38,189,68,124,23,234,113,164,53,204,83,139)(9,238,99,168,39,208,69,143,24,223,114,153,54,193,84,128)(10,227,100,157,40,197,70,132,25,212,115,172,55,182,85,147)(11,216,101,176,41,186,71,121,26,231,116,161,56,201,86,136)(12,235,102,165,42,205,72,140,27,220,117,180,57,190,87,125)(13,224,103,154,43,194,73,129,28,239,118,169,58,209,88,144)(14,213,104,173,44,183,74,148,29,228,119,158,59,198,89,133)(15,232,105,162,45,202,75,137,30,217,120,177,60,187,90,122) );
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),(61,90),(62,89),(63,88),(64,87),(65,86),(66,85),(67,84),(68,83),(69,82),(70,81),(71,80),(72,79),(73,78),(74,77),(75,76),(91,120),(92,119),(93,118),(94,117),(95,116),(96,115),(97,114),(98,113),(99,112),(100,111),(101,110),(102,109),(103,108),(104,107),(105,106),(121,145),(122,144),(123,143),(124,142),(125,141),(126,140),(127,139),(128,138),(129,137),(130,136),(131,135),(132,134),(146,150),(147,149),(151,165),(152,164),(153,163),(154,162),(155,161),(156,160),(157,159),(166,180),(167,179),(168,178),(169,177),(170,176),(171,175),(172,174),(181,185),(182,184),(186,210),(187,209),(188,208),(189,207),(190,206),(191,205),(192,204),(193,203),(194,202),(195,201),(196,200),(197,199),(211,215),(212,214),(216,240),(217,239),(218,238),(219,237),(220,236),(221,235),(222,234),(223,233),(224,232),(225,231),(226,230),(227,229)], [(1,236,91,166,31,206,61,141,16,221,106,151,46,191,76,126),(2,225,92,155,32,195,62,130,17,240,107,170,47,210,77,145),(3,214,93,174,33,184,63,149,18,229,108,159,48,199,78,134),(4,233,94,163,34,203,64,138,19,218,109,178,49,188,79,123),(5,222,95,152,35,192,65,127,20,237,110,167,50,207,80,142),(6,211,96,171,36,181,66,146,21,226,111,156,51,196,81,131),(7,230,97,160,37,200,67,135,22,215,112,175,52,185,82,150),(8,219,98,179,38,189,68,124,23,234,113,164,53,204,83,139),(9,238,99,168,39,208,69,143,24,223,114,153,54,193,84,128),(10,227,100,157,40,197,70,132,25,212,115,172,55,182,85,147),(11,216,101,176,41,186,71,121,26,231,116,161,56,201,86,136),(12,235,102,165,42,205,72,140,27,220,117,180,57,190,87,125),(13,224,103,154,43,194,73,129,28,239,118,169,58,209,88,144),(14,213,104,173,44,183,74,148,29,228,119,158,59,198,89,133),(15,232,105,162,45,202,75,137,30,217,120,177,60,187,90,122)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5A | 5B | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 12A | 12B | 15A | 15B | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 20C | 20D | 24A | 24B | 24C | 24D | 30A | 30B | 40A | ··· | 40H | 48A | ··· | 48H | 60A | 60B | 60C | 60D | 80A | ··· | 80P | 120A | ··· | 120H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 12 | 12 | 15 | 15 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | 30 | 40 | ··· | 40 | 48 | ··· | 48 | 60 | 60 | 60 | 60 | 80 | ··· | 80 | 120 | ··· | 120 |
| size | 1 | 1 | 30 | 2 | 1 | 1 | 30 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | D5 | D6 | D10 | C4×S3 | M5(2) | C4×D5 | S3×C8 | C8×D5 | D6.C8 | C80⋊C2 | S3×D5 | D30.C2 | D15⋊2C8 | D30.5C8 |
| kernel | D30.5C8 | C5×C3⋊C16 | C3×C5⋊2C16 | C8×D15 | C15⋊3C8 | C4×D15 | Dic15 | D30 | C5⋊2C16 | C3⋊C16 | C40 | C24 | C20 | C15 | C12 | C10 | C6 | C5 | C3 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 16 | 2 | 2 | 4 | 8 |
Matrix representation of D30.5C8 ►in GL4(𝔽241) generated by
| 190 | 1 | 0 | 0 |
| 240 | 0 | 0 | 0 |
| 0 | 0 | 240 | 1 |
| 0 | 0 | 240 | 0 |
| 190 | 1 | 0 | 0 |
| 51 | 51 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 233 | 109 | 0 | 0 |
| 183 | 8 | 0 | 0 |
| 0 | 0 | 177 | 0 |
| 0 | 0 | 0 | 177 |
G:=sub<GL(4,GF(241))| [190,240,0,0,1,0,0,0,0,0,240,240,0,0,1,0],[190,51,0,0,1,51,0,0,0,0,0,1,0,0,1,0],[233,183,0,0,109,8,0,0,0,0,177,0,0,0,0,177] >;
D30.5C8 in GAP, Magma, Sage, TeX
D_{30}._5C_8 % in TeX
G:=Group("D30.5C8"); // GroupNames label
G:=SmallGroup(480,12);
// by ID
G=gap.SmallGroup(480,12);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,253,36,58,80,1356,18822]);
// 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^19,c*b*c^-1=a^3*b>;
// generators/relations
Export