metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30⋊7Q8, C60.22D4, (C6×Q8)⋊3D5, (Q8×C30)⋊3C2, (Q8×C10)⋊7S3, (C2×Q8)⋊5D15, C6.46(Q8×D5), C2.9(Q8×D15), C5⋊5(D6⋊3Q8), C60⋊5C4⋊21C2, (C2×C4).56D30, C30.99(C2×Q8), C10.46(S3×Q8), C3⋊5(D10⋊3Q8), C30.389(C2×D4), (C2×C20).155D6, C15⋊37(C22⋊Q8), (C2×C12).251D10, C12.49(C5⋊D4), C4.18(C15⋊7D4), C20.47(C3⋊D4), C30.4Q8⋊38C2, D30⋊3C4.16C2, C30.261(C4○D4), C2.8(Q8⋊3D15), (C2×C60).435C22, (C2×C30).314C23, C6.45(Q8⋊2D5), C10.45(Q8⋊3S3), C22.64(C22×D15), (C2×Dic15).20C22, (C22×D15).89C22, (C2×C4×D15).4C2, C2.21(C2×C15⋊7D4), C6.116(C2×C5⋊D4), C10.116(C2×C3⋊D4), (C2×C6).310(C22×D5), (C2×C10).309(C22×S3), SmallGroup(480,911)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D30⋊7Q8
G = < a,b,c,d | a30=b2=c4=1, d2=c2, bab=a-1, ac=ca, ad=da, cbc-1=a15b, bd=db, dcd-1=c-1 >
Subgroups: 820 in 148 conjugacy classes, 57 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, C2×C4, Q8, C23, D5, C10, Dic3, C12, C12, D6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C4×S3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, D15, C30, C22⋊Q8, C4×D5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, Dic3⋊C4, C4⋊Dic3, D6⋊C4, S3×C2×C4, C6×Q8, Dic15, C60, C60, D30, D30, C2×C30, C10.D4, C4⋊Dic5, D10⋊C4, C2×C4×D5, Q8×C10, D6⋊3Q8, C4×D15, C2×Dic15, C2×Dic15, C2×C60, C2×C60, Q8×C15, C22×D15, D10⋊3Q8, C30.4Q8, C60⋊5C4, D30⋊3C4, C2×C4×D15, Q8×C30, D30⋊7Q8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, C3⋊D4, C22×S3, D15, C22⋊Q8, C5⋊D4, C22×D5, S3×Q8, Q8⋊3S3, C2×C3⋊D4, D30, Q8×D5, Q8⋊2D5, C2×C5⋊D4, D6⋊3Q8, C15⋊7D4, C22×D15, D10⋊3Q8, Q8×D15, Q8⋊3D15, C2×C15⋊7D4, D30⋊7Q8
►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 90)(2 89)(3 88)(4 87)(5 86)(6 85)(7 84)(8 83)(9 82)(10 81)(11 80)(12 79)(13 78)(14 77)(15 76)(16 75)(17 74)(18 73)(19 72)(20 71)(21 70)(22 69)(23 68)(24 67)(25 66)(26 65)(27 64)(28 63)(29 62)(30 61)(31 229)(32 228)(33 227)(34 226)(35 225)(36 224)(37 223)(38 222)(39 221)(40 220)(41 219)(42 218)(43 217)(44 216)(45 215)(46 214)(47 213)(48 212)(49 211)(50 240)(51 239)(52 238)(53 237)(54 236)(55 235)(56 234)(57 233)(58 232)(59 231)(60 230)(91 209)(92 208)(93 207)(94 206)(95 205)(96 204)(97 203)(98 202)(99 201)(100 200)(101 199)(102 198)(103 197)(104 196)(105 195)(106 194)(107 193)(108 192)(109 191)(110 190)(111 189)(112 188)(113 187)(114 186)(115 185)(116 184)(117 183)(118 182)(119 181)(120 210)(121 157)(122 156)(123 155)(124 154)(125 153)(126 152)(127 151)(128 180)(129 179)(130 178)(131 177)(132 176)(133 175)(134 174)(135 173)(136 172)(137 171)(138 170)(139 169)(140 168)(141 167)(142 166)(143 165)(144 164)(145 163)(146 162)(147 161)(148 160)(149 159)(150 158)
(1 225 61 51)(2 226 62 52)(3 227 63 53)(4 228 64 54)(5 229 65 55)(6 230 66 56)(7 231 67 57)(8 232 68 58)(9 233 69 59)(10 234 70 60)(11 235 71 31)(12 236 72 32)(13 237 73 33)(14 238 74 34)(15 239 75 35)(16 240 76 36)(17 211 77 37)(18 212 78 38)(19 213 79 39)(20 214 80 40)(21 215 81 41)(22 216 82 42)(23 217 83 43)(24 218 84 44)(25 219 85 45)(26 220 86 46)(27 221 87 47)(28 222 88 48)(29 223 89 49)(30 224 90 50)(91 167 202 149)(92 168 203 150)(93 169 204 121)(94 170 205 122)(95 171 206 123)(96 172 207 124)(97 173 208 125)(98 174 209 126)(99 175 210 127)(100 176 181 128)(101 177 182 129)(102 178 183 130)(103 179 184 131)(104 180 185 132)(105 151 186 133)(106 152 187 134)(107 153 188 135)(108 154 189 136)(109 155 190 137)(110 156 191 138)(111 157 192 139)(112 158 193 140)(113 159 194 141)(114 160 195 142)(115 161 196 143)(116 162 197 144)(117 163 198 145)(118 164 199 146)(119 165 200 147)(120 166 201 148)
(1 110 61 191)(2 111 62 192)(3 112 63 193)(4 113 64 194)(5 114 65 195)(6 115 66 196)(7 116 67 197)(8 117 68 198)(9 118 69 199)(10 119 70 200)(11 120 71 201)(12 91 72 202)(13 92 73 203)(14 93 74 204)(15 94 75 205)(16 95 76 206)(17 96 77 207)(18 97 78 208)(19 98 79 209)(20 99 80 210)(21 100 81 181)(22 101 82 182)(23 102 83 183)(24 103 84 184)(25 104 85 185)(26 105 86 186)(27 106 87 187)(28 107 88 188)(29 108 89 189)(30 109 90 190)(31 166 235 148)(32 167 236 149)(33 168 237 150)(34 169 238 121)(35 170 239 122)(36 171 240 123)(37 172 211 124)(38 173 212 125)(39 174 213 126)(40 175 214 127)(41 176 215 128)(42 177 216 129)(43 178 217 130)(44 179 218 131)(45 180 219 132)(46 151 220 133)(47 152 221 134)(48 153 222 135)(49 154 223 136)(50 155 224 137)(51 156 225 138)(52 157 226 139)(53 158 227 140)(54 159 228 141)(55 160 229 142)(56 161 230 143)(57 162 231 144)(58 163 232 145)(59 164 233 146)(60 165 234 147)
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), (1,90)(2,89)(3,88)(4,87)(5,86)(6,85)(7,84)(8,83)(9,82)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,229)(32,228)(33,227)(34,226)(35,225)(36,224)(37,223)(38,222)(39,221)(40,220)(41,219)(42,218)(43,217)(44,216)(45,215)(46,214)(47,213)(48,212)(49,211)(50,240)(51,239)(52,238)(53,237)(54,236)(55,235)(56,234)(57,233)(58,232)(59,231)(60,230)(91,209)(92,208)(93,207)(94,206)(95,205)(96,204)(97,203)(98,202)(99,201)(100,200)(101,199)(102,198)(103,197)(104,196)(105,195)(106,194)(107,193)(108,192)(109,191)(110,190)(111,189)(112,188)(113,187)(114,186)(115,185)(116,184)(117,183)(118,182)(119,181)(120,210)(121,157)(122,156)(123,155)(124,154)(125,153)(126,152)(127,151)(128,180)(129,179)(130,178)(131,177)(132,176)(133,175)(134,174)(135,173)(136,172)(137,171)(138,170)(139,169)(140,168)(141,167)(142,166)(143,165)(144,164)(145,163)(146,162)(147,161)(148,160)(149,159)(150,158), (1,225,61,51)(2,226,62,52)(3,227,63,53)(4,228,64,54)(5,229,65,55)(6,230,66,56)(7,231,67,57)(8,232,68,58)(9,233,69,59)(10,234,70,60)(11,235,71,31)(12,236,72,32)(13,237,73,33)(14,238,74,34)(15,239,75,35)(16,240,76,36)(17,211,77,37)(18,212,78,38)(19,213,79,39)(20,214,80,40)(21,215,81,41)(22,216,82,42)(23,217,83,43)(24,218,84,44)(25,219,85,45)(26,220,86,46)(27,221,87,47)(28,222,88,48)(29,223,89,49)(30,224,90,50)(91,167,202,149)(92,168,203,150)(93,169,204,121)(94,170,205,122)(95,171,206,123)(96,172,207,124)(97,173,208,125)(98,174,209,126)(99,175,210,127)(100,176,181,128)(101,177,182,129)(102,178,183,130)(103,179,184,131)(104,180,185,132)(105,151,186,133)(106,152,187,134)(107,153,188,135)(108,154,189,136)(109,155,190,137)(110,156,191,138)(111,157,192,139)(112,158,193,140)(113,159,194,141)(114,160,195,142)(115,161,196,143)(116,162,197,144)(117,163,198,145)(118,164,199,146)(119,165,200,147)(120,166,201,148), (1,110,61,191)(2,111,62,192)(3,112,63,193)(4,113,64,194)(5,114,65,195)(6,115,66,196)(7,116,67,197)(8,117,68,198)(9,118,69,199)(10,119,70,200)(11,120,71,201)(12,91,72,202)(13,92,73,203)(14,93,74,204)(15,94,75,205)(16,95,76,206)(17,96,77,207)(18,97,78,208)(19,98,79,209)(20,99,80,210)(21,100,81,181)(22,101,82,182)(23,102,83,183)(24,103,84,184)(25,104,85,185)(26,105,86,186)(27,106,87,187)(28,107,88,188)(29,108,89,189)(30,109,90,190)(31,166,235,148)(32,167,236,149)(33,168,237,150)(34,169,238,121)(35,170,239,122)(36,171,240,123)(37,172,211,124)(38,173,212,125)(39,174,213,126)(40,175,214,127)(41,176,215,128)(42,177,216,129)(43,178,217,130)(44,179,218,131)(45,180,219,132)(46,151,220,133)(47,152,221,134)(48,153,222,135)(49,154,223,136)(50,155,224,137)(51,156,225,138)(52,157,226,139)(53,158,227,140)(54,159,228,141)(55,160,229,142)(56,161,230,143)(57,162,231,144)(58,163,232,145)(59,164,233,146)(60,165,234,147)>;
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), (1,90)(2,89)(3,88)(4,87)(5,86)(6,85)(7,84)(8,83)(9,82)(10,81)(11,80)(12,79)(13,78)(14,77)(15,76)(16,75)(17,74)(18,73)(19,72)(20,71)(21,70)(22,69)(23,68)(24,67)(25,66)(26,65)(27,64)(28,63)(29,62)(30,61)(31,229)(32,228)(33,227)(34,226)(35,225)(36,224)(37,223)(38,222)(39,221)(40,220)(41,219)(42,218)(43,217)(44,216)(45,215)(46,214)(47,213)(48,212)(49,211)(50,240)(51,239)(52,238)(53,237)(54,236)(55,235)(56,234)(57,233)(58,232)(59,231)(60,230)(91,209)(92,208)(93,207)(94,206)(95,205)(96,204)(97,203)(98,202)(99,201)(100,200)(101,199)(102,198)(103,197)(104,196)(105,195)(106,194)(107,193)(108,192)(109,191)(110,190)(111,189)(112,188)(113,187)(114,186)(115,185)(116,184)(117,183)(118,182)(119,181)(120,210)(121,157)(122,156)(123,155)(124,154)(125,153)(126,152)(127,151)(128,180)(129,179)(130,178)(131,177)(132,176)(133,175)(134,174)(135,173)(136,172)(137,171)(138,170)(139,169)(140,168)(141,167)(142,166)(143,165)(144,164)(145,163)(146,162)(147,161)(148,160)(149,159)(150,158), (1,225,61,51)(2,226,62,52)(3,227,63,53)(4,228,64,54)(5,229,65,55)(6,230,66,56)(7,231,67,57)(8,232,68,58)(9,233,69,59)(10,234,70,60)(11,235,71,31)(12,236,72,32)(13,237,73,33)(14,238,74,34)(15,239,75,35)(16,240,76,36)(17,211,77,37)(18,212,78,38)(19,213,79,39)(20,214,80,40)(21,215,81,41)(22,216,82,42)(23,217,83,43)(24,218,84,44)(25,219,85,45)(26,220,86,46)(27,221,87,47)(28,222,88,48)(29,223,89,49)(30,224,90,50)(91,167,202,149)(92,168,203,150)(93,169,204,121)(94,170,205,122)(95,171,206,123)(96,172,207,124)(97,173,208,125)(98,174,209,126)(99,175,210,127)(100,176,181,128)(101,177,182,129)(102,178,183,130)(103,179,184,131)(104,180,185,132)(105,151,186,133)(106,152,187,134)(107,153,188,135)(108,154,189,136)(109,155,190,137)(110,156,191,138)(111,157,192,139)(112,158,193,140)(113,159,194,141)(114,160,195,142)(115,161,196,143)(116,162,197,144)(117,163,198,145)(118,164,199,146)(119,165,200,147)(120,166,201,148), (1,110,61,191)(2,111,62,192)(3,112,63,193)(4,113,64,194)(5,114,65,195)(6,115,66,196)(7,116,67,197)(8,117,68,198)(9,118,69,199)(10,119,70,200)(11,120,71,201)(12,91,72,202)(13,92,73,203)(14,93,74,204)(15,94,75,205)(16,95,76,206)(17,96,77,207)(18,97,78,208)(19,98,79,209)(20,99,80,210)(21,100,81,181)(22,101,82,182)(23,102,83,183)(24,103,84,184)(25,104,85,185)(26,105,86,186)(27,106,87,187)(28,107,88,188)(29,108,89,189)(30,109,90,190)(31,166,235,148)(32,167,236,149)(33,168,237,150)(34,169,238,121)(35,170,239,122)(36,171,240,123)(37,172,211,124)(38,173,212,125)(39,174,213,126)(40,175,214,127)(41,176,215,128)(42,177,216,129)(43,178,217,130)(44,179,218,131)(45,180,219,132)(46,151,220,133)(47,152,221,134)(48,153,222,135)(49,154,223,136)(50,155,224,137)(51,156,225,138)(52,157,226,139)(53,158,227,140)(54,159,228,141)(55,160,229,142)(56,161,230,143)(57,162,231,144)(58,163,232,145)(59,164,233,146)(60,165,234,147) );
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)], [(1,90),(2,89),(3,88),(4,87),(5,86),(6,85),(7,84),(8,83),(9,82),(10,81),(11,80),(12,79),(13,78),(14,77),(15,76),(16,75),(17,74),(18,73),(19,72),(20,71),(21,70),(22,69),(23,68),(24,67),(25,66),(26,65),(27,64),(28,63),(29,62),(30,61),(31,229),(32,228),(33,227),(34,226),(35,225),(36,224),(37,223),(38,222),(39,221),(40,220),(41,219),(42,218),(43,217),(44,216),(45,215),(46,214),(47,213),(48,212),(49,211),(50,240),(51,239),(52,238),(53,237),(54,236),(55,235),(56,234),(57,233),(58,232),(59,231),(60,230),(91,209),(92,208),(93,207),(94,206),(95,205),(96,204),(97,203),(98,202),(99,201),(100,200),(101,199),(102,198),(103,197),(104,196),(105,195),(106,194),(107,193),(108,192),(109,191),(110,190),(111,189),(112,188),(113,187),(114,186),(115,185),(116,184),(117,183),(118,182),(119,181),(120,210),(121,157),(122,156),(123,155),(124,154),(125,153),(126,152),(127,151),(128,180),(129,179),(130,178),(131,177),(132,176),(133,175),(134,174),(135,173),(136,172),(137,171),(138,170),(139,169),(140,168),(141,167),(142,166),(143,165),(144,164),(145,163),(146,162),(147,161),(148,160),(149,159),(150,158)], [(1,225,61,51),(2,226,62,52),(3,227,63,53),(4,228,64,54),(5,229,65,55),(6,230,66,56),(7,231,67,57),(8,232,68,58),(9,233,69,59),(10,234,70,60),(11,235,71,31),(12,236,72,32),(13,237,73,33),(14,238,74,34),(15,239,75,35),(16,240,76,36),(17,211,77,37),(18,212,78,38),(19,213,79,39),(20,214,80,40),(21,215,81,41),(22,216,82,42),(23,217,83,43),(24,218,84,44),(25,219,85,45),(26,220,86,46),(27,221,87,47),(28,222,88,48),(29,223,89,49),(30,224,90,50),(91,167,202,149),(92,168,203,150),(93,169,204,121),(94,170,205,122),(95,171,206,123),(96,172,207,124),(97,173,208,125),(98,174,209,126),(99,175,210,127),(100,176,181,128),(101,177,182,129),(102,178,183,130),(103,179,184,131),(104,180,185,132),(105,151,186,133),(106,152,187,134),(107,153,188,135),(108,154,189,136),(109,155,190,137),(110,156,191,138),(111,157,192,139),(112,158,193,140),(113,159,194,141),(114,160,195,142),(115,161,196,143),(116,162,197,144),(117,163,198,145),(118,164,199,146),(119,165,200,147),(120,166,201,148)], [(1,110,61,191),(2,111,62,192),(3,112,63,193),(4,113,64,194),(5,114,65,195),(6,115,66,196),(7,116,67,197),(8,117,68,198),(9,118,69,199),(10,119,70,200),(11,120,71,201),(12,91,72,202),(13,92,73,203),(14,93,74,204),(15,94,75,205),(16,95,76,206),(17,96,77,207),(18,97,78,208),(19,98,79,209),(20,99,80,210),(21,100,81,181),(22,101,82,182),(23,102,83,183),(24,103,84,184),(25,104,85,185),(26,105,86,186),(27,106,87,187),(28,107,88,188),(29,108,89,189),(30,109,90,190),(31,166,235,148),(32,167,236,149),(33,168,237,150),(34,169,238,121),(35,170,239,122),(36,171,240,123),(37,172,211,124),(38,173,212,125),(39,174,213,126),(40,175,214,127),(41,176,215,128),(42,177,216,129),(43,178,217,130),(44,179,218,131),(45,180,219,132),(46,151,220,133),(47,152,221,134),(48,153,222,135),(49,154,223,136),(50,155,224,137),(51,156,225,138),(52,157,226,139),(53,158,227,140),(54,159,228,141),(55,160,229,142),(56,161,230,143),(57,162,231,144),(58,163,232,145),(59,164,233,146),(60,165,234,147)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 2 | 2 | 2 | 4 | 4 | 30 | 30 | 60 | 60 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | + | - | + | - | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D5 | D6 | C4○D4 | D10 | C3⋊D4 | D15 | C5⋊D4 | D30 | C15⋊7D4 | S3×Q8 | Q8⋊3S3 | Q8×D5 | Q8⋊2D5 | Q8×D15 | Q8⋊3D15 |
| kernel | D30⋊7Q8 | C30.4Q8 | C60⋊5C4 | D30⋊3C4 | C2×C4×D15 | Q8×C30 | Q8×C10 | C60 | D30 | C6×Q8 | C2×C20 | C30 | C2×C12 | C20 | C2×Q8 | C12 | C2×C4 | C4 | C10 | C10 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 3 | 2 | 6 | 4 | 4 | 8 | 12 | 16 | 1 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of D30⋊7Q8 ►in GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 25 | 0 | 0 |
| 0 | 0 | 13 | 53 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 55 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 17 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 27 | 52 | 0 | 0 | 0 | 0 |
| 47 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 50 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,5,13,0,0,0,0,25,53,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,55,0,0,0,0,0,1,0,0,0,0,0,0,0,18,0,0,0,0,17,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60],[27,47,0,0,0,0,52,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,0,0,0,0,0,0,50] >;
D30⋊7Q8 in GAP, Magma, Sage, TeX
D_{30}\rtimes_7Q_8 % in TeX
G:=Group("D30:7Q8"); // GroupNames label
G:=SmallGroup(480,911);
// by ID
G=gap.SmallGroup(480,911);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,254,219,100,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^30=b^2=c^4=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations