metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊4C8, C60.95D4, C20.55D12, C12.55D20, C30.16M4(2), C5⋊3(D6⋊C8), C6.4(C8×D5), C10.13(S3×C8), C15⋊7(C22⋊C8), C30.30(C2×C8), C3⋊1(D10⋊1C8), (C2×C20).323D6, C6.2(C8⋊D5), C10.7(C8⋊S3), C10.16(D6⋊C4), (C2×C12).327D10, C20.60(C3⋊D4), C4.27(C5⋊D12), C4.27(C3⋊D20), C12.63(C5⋊D4), C2.1(D30⋊4C4), C2.4(D15⋊2C8), C6.1(D10⋊C4), C30.40(C22⋊C4), (C2×C60).225C22, (C2×Dic15).19C4, (C22×D15).11C4, C2.2(D30.5C4), C22.10(D30.C2), (C2×C3⋊C8)⋊10D5, (C10×C3⋊C8)⋊10C2, (C6×C5⋊2C8)⋊10C2, (C2×C5⋊2C8)⋊10S3, (C2×C4×D15).19C2, (C2×C6).18(C4×D5), (C2×C10).41(C4×S3), (C2×C30).79(C2×C4), (C2×C4).228(S3×D5), SmallGroup(480,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊4C8
G = < a,b,c | a30=b2=c8=1, bab=a-1, cac-1=a11, cbc-1=a25b >
Subgroups: 572 in 100 conjugacy classes, 42 normal (40 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, 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, C5×C3⋊C8, C3×C5⋊2C8, C4×D15, C2×Dic15, C2×C60, C22×D15, D10⋊1C8, C6×C5⋊2C8, C10×C3⋊C8, C2×C4×D15, D30⋊4C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, D5, D6, C22⋊C4, C2×C8, M4(2), D10, C4×S3, D12, C3⋊D4, C22⋊C8, C4×D5, D20, C5⋊D4, S3×C8, C8⋊S3, D6⋊C4, S3×D5, C8×D5, C8⋊D5, D10⋊C4, D6⋊C8, D30.C2, C3⋊D20, C5⋊D12, D10⋊1C8, D15⋊2C8, D30.5C4, D30⋊4C4, D30⋊4C8
►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 42)(32 41)(33 40)(34 39)(35 38)(36 37)(43 60)(44 59)(45 58)(46 57)(47 56)(48 55)(49 54)(50 53)(51 52)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(75 90)(76 89)(77 88)(78 87)(79 86)(80 85)(81 84)(82 83)(91 95)(92 94)(96 120)(97 119)(98 118)(99 117)(100 116)(101 115)(102 114)(103 113)(104 112)(105 111)(106 110)(107 109)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(136 150)(137 149)(138 148)(139 147)(140 146)(141 145)(142 144)(151 173)(152 172)(153 171)(154 170)(155 169)(156 168)(157 167)(158 166)(159 165)(160 164)(161 163)(174 180)(175 179)(176 178)(181 194)(182 193)(183 192)(184 191)(185 190)(186 189)(187 188)(195 210)(196 209)(197 208)(198 207)(199 206)(200 205)(201 204)(202 203)(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 68 101 37 170 188 121)(2 217 69 112 38 151 189 132)(3 228 70 93 39 162 190 143)(4 239 71 104 40 173 191 124)(5 220 72 115 41 154 192 135)(6 231 73 96 42 165 193 146)(7 212 74 107 43 176 194 127)(8 223 75 118 44 157 195 138)(9 234 76 99 45 168 196 149)(10 215 77 110 46 179 197 130)(11 226 78 91 47 160 198 141)(12 237 79 102 48 171 199 122)(13 218 80 113 49 152 200 133)(14 229 81 94 50 163 201 144)(15 240 82 105 51 174 202 125)(16 221 83 116 52 155 203 136)(17 232 84 97 53 166 204 147)(18 213 85 108 54 177 205 128)(19 224 86 119 55 158 206 139)(20 235 87 100 56 169 207 150)(21 216 88 111 57 180 208 131)(22 227 89 92 58 161 209 142)(23 238 90 103 59 172 210 123)(24 219 61 114 60 153 181 134)(25 230 62 95 31 164 182 145)(26 211 63 106 32 175 183 126)(27 222 64 117 33 156 184 137)(28 233 65 98 34 167 185 148)(29 214 66 109 35 178 186 129)(30 225 67 120 36 159 187 140)
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)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)(91,95)(92,94)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,150)(137,149)(138,148)(139,147)(140,146)(141,145)(142,144)(151,173)(152,172)(153,171)(154,170)(155,169)(156,168)(157,167)(158,166)(159,165)(160,164)(161,163)(174,180)(175,179)(176,178)(181,194)(182,193)(183,192)(184,191)(185,190)(186,189)(187,188)(195,210)(196,209)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)(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), (1,236,68,101,37,170,188,121)(2,217,69,112,38,151,189,132)(3,228,70,93,39,162,190,143)(4,239,71,104,40,173,191,124)(5,220,72,115,41,154,192,135)(6,231,73,96,42,165,193,146)(7,212,74,107,43,176,194,127)(8,223,75,118,44,157,195,138)(9,234,76,99,45,168,196,149)(10,215,77,110,46,179,197,130)(11,226,78,91,47,160,198,141)(12,237,79,102,48,171,199,122)(13,218,80,113,49,152,200,133)(14,229,81,94,50,163,201,144)(15,240,82,105,51,174,202,125)(16,221,83,116,52,155,203,136)(17,232,84,97,53,166,204,147)(18,213,85,108,54,177,205,128)(19,224,86,119,55,158,206,139)(20,235,87,100,56,169,207,150)(21,216,88,111,57,180,208,131)(22,227,89,92,58,161,209,142)(23,238,90,103,59,172,210,123)(24,219,61,114,60,153,181,134)(25,230,62,95,31,164,182,145)(26,211,63,106,32,175,183,126)(27,222,64,117,33,156,184,137)(28,233,65,98,34,167,185,148)(29,214,66,109,35,178,186,129)(30,225,67,120,36,159,187,140)>;
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)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)(75,90)(76,89)(77,88)(78,87)(79,86)(80,85)(81,84)(82,83)(91,95)(92,94)(96,120)(97,119)(98,118)(99,117)(100,116)(101,115)(102,114)(103,113)(104,112)(105,111)(106,110)(107,109)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,150)(137,149)(138,148)(139,147)(140,146)(141,145)(142,144)(151,173)(152,172)(153,171)(154,170)(155,169)(156,168)(157,167)(158,166)(159,165)(160,164)(161,163)(174,180)(175,179)(176,178)(181,194)(182,193)(183,192)(184,191)(185,190)(186,189)(187,188)(195,210)(196,209)(197,208)(198,207)(199,206)(200,205)(201,204)(202,203)(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), (1,236,68,101,37,170,188,121)(2,217,69,112,38,151,189,132)(3,228,70,93,39,162,190,143)(4,239,71,104,40,173,191,124)(5,220,72,115,41,154,192,135)(6,231,73,96,42,165,193,146)(7,212,74,107,43,176,194,127)(8,223,75,118,44,157,195,138)(9,234,76,99,45,168,196,149)(10,215,77,110,46,179,197,130)(11,226,78,91,47,160,198,141)(12,237,79,102,48,171,199,122)(13,218,80,113,49,152,200,133)(14,229,81,94,50,163,201,144)(15,240,82,105,51,174,202,125)(16,221,83,116,52,155,203,136)(17,232,84,97,53,166,204,147)(18,213,85,108,54,177,205,128)(19,224,86,119,55,158,206,139)(20,235,87,100,56,169,207,150)(21,216,88,111,57,180,208,131)(22,227,89,92,58,161,209,142)(23,238,90,103,59,172,210,123)(24,219,61,114,60,153,181,134)(25,230,62,95,31,164,182,145)(26,211,63,106,32,175,183,126)(27,222,64,117,33,156,184,137)(28,233,65,98,34,167,185,148)(29,214,66,109,35,178,186,129)(30,225,67,120,36,159,187,140) );
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),(31,42),(32,41),(33,40),(34,39),(35,38),(36,37),(43,60),(44,59),(45,58),(46,57),(47,56),(48,55),(49,54),(50,53),(51,52),(61,74),(62,73),(63,72),(64,71),(65,70),(66,69),(67,68),(75,90),(76,89),(77,88),(78,87),(79,86),(80,85),(81,84),(82,83),(91,95),(92,94),(96,120),(97,119),(98,118),(99,117),(100,116),(101,115),(102,114),(103,113),(104,112),(105,111),(106,110),(107,109),(121,135),(122,134),(123,133),(124,132),(125,131),(126,130),(127,129),(136,150),(137,149),(138,148),(139,147),(140,146),(141,145),(142,144),(151,173),(152,172),(153,171),(154,170),(155,169),(156,168),(157,167),(158,166),(159,165),(160,164),(161,163),(174,180),(175,179),(176,178),(181,194),(182,193),(183,192),(184,191),(185,190),(186,189),(187,188),(195,210),(196,209),(197,208),(198,207),(199,206),(200,205),(201,204),(202,203),(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)], [(1,236,68,101,37,170,188,121),(2,217,69,112,38,151,189,132),(3,228,70,93,39,162,190,143),(4,239,71,104,40,173,191,124),(5,220,72,115,41,154,192,135),(6,231,73,96,42,165,193,146),(7,212,74,107,43,176,194,127),(8,223,75,118,44,157,195,138),(9,234,76,99,45,168,196,149),(10,215,77,110,46,179,197,130),(11,226,78,91,47,160,198,141),(12,237,79,102,48,171,199,122),(13,218,80,113,49,152,200,133),(14,229,81,94,50,163,201,144),(15,240,82,105,51,174,202,125),(16,221,83,116,52,155,203,136),(17,232,84,97,53,166,204,147),(18,213,85,108,54,177,205,128),(19,224,86,119,55,158,206,139),(20,235,87,100,56,169,207,150),(21,216,88,111,57,180,208,131),(22,227,89,92,58,161,209,142),(23,238,90,103,59,172,210,123),(24,219,61,114,60,153,181,134),(25,230,62,95,31,164,182,145),(26,211,63,106,32,175,183,126),(27,222,64,117,33,156,184,137),(28,233,65,98,34,167,185,148),(29,214,66,109,35,178,186,129),(30,225,67,120,36,159,187,140)]])
84 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 | 20A | ··· | 20H | 24A | ··· | 24H | 30A | ··· | 30F | 40A | ··· | 40P | 60A | ··· | 60H |
| 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 | 20 | ··· | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | ··· | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
84 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 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | D5 | D6 | M4(2) | D10 | D12 | C3⋊D4 | C4×S3 | D20 | C5⋊D4 | C4×D5 | S3×C8 | C8⋊S3 | C8×D5 | C8⋊D5 | S3×D5 | C3⋊D20 | C5⋊D12 | D30.C2 | D15⋊2C8 | D30.5C4 |
| kernel | D30⋊4C8 | C6×C5⋊2C8 | C10×C3⋊C8 | C2×C4×D15 | C2×Dic15 | C22×D15 | D30 | C2×C5⋊2C8 | C60 | C2×C3⋊C8 | C2×C20 | C30 | C2×C12 | C20 | C20 | C2×C10 | C12 | C12 | C2×C6 | C10 | C10 | C6 | C6 | C2×C4 | 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 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of D30⋊4C8 ►in GL5(𝔽241)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 52 | 1 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 240 | 240 |
| 240 | 0 | 0 | 0 | 0 |
| 0 | 52 | 1 | 0 | 0 |
| 0 | 189 | 189 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 30 | 0 | 0 | 0 | 0 |
| 0 | 87 | 198 | 0 | 0 |
| 0 | 43 | 154 | 0 | 0 |
| 0 | 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 177 | 177 |
G:=sub<GL(5,GF(241))| [1,0,0,0,0,0,52,240,0,0,0,1,0,0,0,0,0,0,0,240,0,0,0,1,240],[240,0,0,0,0,0,52,189,0,0,0,1,189,0,0,0,0,0,0,1,0,0,0,1,0],[30,0,0,0,0,0,87,43,0,0,0,198,154,0,0,0,0,0,64,177,0,0,0,0,177] >;
D30⋊4C8 in GAP, Magma, Sage, TeX
D_{30}\rtimes_4C_8 % in TeX
G:=Group("D30:4C8"); // GroupNames label
G:=SmallGroup(480,33);
// by ID
G=gap.SmallGroup(480,33);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,92,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^30=b^2=c^8=1,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^25*b>;
// generators/relations