metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.15D30, C60.87C23, C30.63C24, C15⋊82- 1+4, D60.42C22, D30.28C23, Dic30.44C22, Dic15.30C23, (C6×Q8)⋊7D5, (C2×Q8)⋊7D15, (Q8×C30)⋊7C2, (Q8×C10)⋊11S3, (Q8×D15)⋊11C2, (C2×C4).23D30, (C5×Q8).53D6, (C2×C20).173D6, (C3×Q8).36D10, Q8⋊3D15⋊11C2, C6.63(C23×D5), (C2×C12).171D10, C10.63(S3×C23), (C2×C60).89C22, D60⋊11C2⋊16C2, C5⋊4(Q8.15D6), C4.24(C22×D15), C2.11(C23×D15), C15⋊7D4.5C22, C20.137(C22×S3), (C2×C30).324C23, C3⋊4(Q8.10D10), (C4×D15).28C22, C12.135(C22×D5), (Q8×C15).41C22, C22.7(C22×D15), (C2×C6).320(C22×D5), (C2×C10).320(C22×S3), SmallGroup(480,1174)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.15D30
G = < a,b,c,d | a4=1, b2=c30=d2=a2, bab-1=cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c29 >
Subgroups: 1492 in 292 conjugacy classes, 119 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, Q8, Q8, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×Q8, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×Q8, D15, C30, C30, 2- 1+4, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×Q8, C4○D12, S3×Q8, Q8⋊3S3, C6×Q8, Dic15, C60, D30, C2×C30, C4○D20, Q8×D5, Q8⋊2D5, Q8×C10, Q8.15D6, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, Q8×C15, Q8.10D10, D60⋊11C2, Q8×D15, Q8⋊3D15, Q8×C30, Q8.15D30
Quotients: C1, C2, C22, S3, C23, D5, D6, C24, D10, C22×S3, D15, 2- 1+4, C22×D5, S3×C23, D30, C23×D5, Q8.15D6, C22×D15, Q8.10D10, C23×D15, Q8.15D30
►On 240 points
Generators in S240
(1 231 31 201)(2 202 32 232)(3 233 33 203)(4 204 34 234)(5 235 35 205)(6 206 36 236)(7 237 37 207)(8 208 38 238)(9 239 39 209)(10 210 40 240)(11 181 41 211)(12 212 42 182)(13 183 43 213)(14 214 44 184)(15 185 45 215)(16 216 46 186)(17 187 47 217)(18 218 48 188)(19 189 49 219)(20 220 50 190)(21 191 51 221)(22 222 52 192)(23 193 53 223)(24 224 54 194)(25 195 55 225)(26 226 56 196)(27 197 57 227)(28 228 58 198)(29 199 59 229)(30 230 60 200)(61 171 91 141)(62 142 92 172)(63 173 93 143)(64 144 94 174)(65 175 95 145)(66 146 96 176)(67 177 97 147)(68 148 98 178)(69 179 99 149)(70 150 100 180)(71 121 101 151)(72 152 102 122)(73 123 103 153)(74 154 104 124)(75 125 105 155)(76 156 106 126)(77 127 107 157)(78 158 108 128)(79 129 109 159)(80 160 110 130)(81 131 111 161)(82 162 112 132)(83 133 113 163)(84 164 114 134)(85 135 115 165)(86 166 116 136)(87 137 117 167)(88 168 118 138)(89 139 119 169)(90 170 120 140)
(1 46 31 16)(2 47 32 17)(3 48 33 18)(4 49 34 19)(5 50 35 20)(6 51 36 21)(7 52 37 22)(8 53 38 23)(9 54 39 24)(10 55 40 25)(11 56 41 26)(12 57 42 27)(13 58 43 28)(14 59 44 29)(15 60 45 30)(61 76 91 106)(62 77 92 107)(63 78 93 108)(64 79 94 109)(65 80 95 110)(66 81 96 111)(67 82 97 112)(68 83 98 113)(69 84 99 114)(70 85 100 115)(71 86 101 116)(72 87 102 117)(73 88 103 118)(74 89 104 119)(75 90 105 120)(121 136 151 166)(122 137 152 167)(123 138 153 168)(124 139 154 169)(125 140 155 170)(126 141 156 171)(127 142 157 172)(128 143 158 173)(129 144 159 174)(130 145 160 175)(131 146 161 176)(132 147 162 177)(133 148 163 178)(134 149 164 179)(135 150 165 180)(181 226 211 196)(182 227 212 197)(183 228 213 198)(184 229 214 199)(185 230 215 200)(186 231 216 201)(187 232 217 202)(188 233 218 203)(189 234 219 204)(190 235 220 205)(191 236 221 206)(192 237 222 207)(193 238 223 208)(194 239 224 209)(195 240 225 210)
(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 88 31 118)(2 117 32 87)(3 86 33 116)(4 115 34 85)(5 84 35 114)(6 113 36 83)(7 82 37 112)(8 111 38 81)(9 80 39 110)(10 109 40 79)(11 78 41 108)(12 107 42 77)(13 76 43 106)(14 105 44 75)(15 74 45 104)(16 103 46 73)(17 72 47 102)(18 101 48 71)(19 70 49 100)(20 99 50 69)(21 68 51 98)(22 97 52 67)(23 66 53 96)(24 95 54 65)(25 64 55 94)(26 93 56 63)(27 62 57 92)(28 91 58 61)(29 120 59 90)(30 89 60 119)(121 188 151 218)(122 217 152 187)(123 186 153 216)(124 215 154 185)(125 184 155 214)(126 213 156 183)(127 182 157 212)(128 211 158 181)(129 240 159 210)(130 209 160 239)(131 238 161 208)(132 207 162 237)(133 236 163 206)(134 205 164 235)(135 234 165 204)(136 203 166 233)(137 232 167 202)(138 201 168 231)(139 230 169 200)(140 199 170 229)(141 228 171 198)(142 197 172 227)(143 226 173 196)(144 195 174 225)(145 224 175 194)(146 193 176 223)(147 222 177 192)(148 191 178 221)(149 220 179 190)(150 189 180 219)
G:=sub<Sym(240)| (1,231,31,201)(2,202,32,232)(3,233,33,203)(4,204,34,234)(5,235,35,205)(6,206,36,236)(7,237,37,207)(8,208,38,238)(9,239,39,209)(10,210,40,240)(11,181,41,211)(12,212,42,182)(13,183,43,213)(14,214,44,184)(15,185,45,215)(16,216,46,186)(17,187,47,217)(18,218,48,188)(19,189,49,219)(20,220,50,190)(21,191,51,221)(22,222,52,192)(23,193,53,223)(24,224,54,194)(25,195,55,225)(26,226,56,196)(27,197,57,227)(28,228,58,198)(29,199,59,229)(30,230,60,200)(61,171,91,141)(62,142,92,172)(63,173,93,143)(64,144,94,174)(65,175,95,145)(66,146,96,176)(67,177,97,147)(68,148,98,178)(69,179,99,149)(70,150,100,180)(71,121,101,151)(72,152,102,122)(73,123,103,153)(74,154,104,124)(75,125,105,155)(76,156,106,126)(77,127,107,157)(78,158,108,128)(79,129,109,159)(80,160,110,130)(81,131,111,161)(82,162,112,132)(83,133,113,163)(84,164,114,134)(85,135,115,165)(86,166,116,136)(87,137,117,167)(88,168,118,138)(89,139,119,169)(90,170,120,140), (1,46,31,16)(2,47,32,17)(3,48,33,18)(4,49,34,19)(5,50,35,20)(6,51,36,21)(7,52,37,22)(8,53,38,23)(9,54,39,24)(10,55,40,25)(11,56,41,26)(12,57,42,27)(13,58,43,28)(14,59,44,29)(15,60,45,30)(61,76,91,106)(62,77,92,107)(63,78,93,108)(64,79,94,109)(65,80,95,110)(66,81,96,111)(67,82,97,112)(68,83,98,113)(69,84,99,114)(70,85,100,115)(71,86,101,116)(72,87,102,117)(73,88,103,118)(74,89,104,119)(75,90,105,120)(121,136,151,166)(122,137,152,167)(123,138,153,168)(124,139,154,169)(125,140,155,170)(126,141,156,171)(127,142,157,172)(128,143,158,173)(129,144,159,174)(130,145,160,175)(131,146,161,176)(132,147,162,177)(133,148,163,178)(134,149,164,179)(135,150,165,180)(181,226,211,196)(182,227,212,197)(183,228,213,198)(184,229,214,199)(185,230,215,200)(186,231,216,201)(187,232,217,202)(188,233,218,203)(189,234,219,204)(190,235,220,205)(191,236,221,206)(192,237,222,207)(193,238,223,208)(194,239,224,209)(195,240,225,210), (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), (1,88,31,118)(2,117,32,87)(3,86,33,116)(4,115,34,85)(5,84,35,114)(6,113,36,83)(7,82,37,112)(8,111,38,81)(9,80,39,110)(10,109,40,79)(11,78,41,108)(12,107,42,77)(13,76,43,106)(14,105,44,75)(15,74,45,104)(16,103,46,73)(17,72,47,102)(18,101,48,71)(19,70,49,100)(20,99,50,69)(21,68,51,98)(22,97,52,67)(23,66,53,96)(24,95,54,65)(25,64,55,94)(26,93,56,63)(27,62,57,92)(28,91,58,61)(29,120,59,90)(30,89,60,119)(121,188,151,218)(122,217,152,187)(123,186,153,216)(124,215,154,185)(125,184,155,214)(126,213,156,183)(127,182,157,212)(128,211,158,181)(129,240,159,210)(130,209,160,239)(131,238,161,208)(132,207,162,237)(133,236,163,206)(134,205,164,235)(135,234,165,204)(136,203,166,233)(137,232,167,202)(138,201,168,231)(139,230,169,200)(140,199,170,229)(141,228,171,198)(142,197,172,227)(143,226,173,196)(144,195,174,225)(145,224,175,194)(146,193,176,223)(147,222,177,192)(148,191,178,221)(149,220,179,190)(150,189,180,219)>;
G:=Group( (1,231,31,201)(2,202,32,232)(3,233,33,203)(4,204,34,234)(5,235,35,205)(6,206,36,236)(7,237,37,207)(8,208,38,238)(9,239,39,209)(10,210,40,240)(11,181,41,211)(12,212,42,182)(13,183,43,213)(14,214,44,184)(15,185,45,215)(16,216,46,186)(17,187,47,217)(18,218,48,188)(19,189,49,219)(20,220,50,190)(21,191,51,221)(22,222,52,192)(23,193,53,223)(24,224,54,194)(25,195,55,225)(26,226,56,196)(27,197,57,227)(28,228,58,198)(29,199,59,229)(30,230,60,200)(61,171,91,141)(62,142,92,172)(63,173,93,143)(64,144,94,174)(65,175,95,145)(66,146,96,176)(67,177,97,147)(68,148,98,178)(69,179,99,149)(70,150,100,180)(71,121,101,151)(72,152,102,122)(73,123,103,153)(74,154,104,124)(75,125,105,155)(76,156,106,126)(77,127,107,157)(78,158,108,128)(79,129,109,159)(80,160,110,130)(81,131,111,161)(82,162,112,132)(83,133,113,163)(84,164,114,134)(85,135,115,165)(86,166,116,136)(87,137,117,167)(88,168,118,138)(89,139,119,169)(90,170,120,140), (1,46,31,16)(2,47,32,17)(3,48,33,18)(4,49,34,19)(5,50,35,20)(6,51,36,21)(7,52,37,22)(8,53,38,23)(9,54,39,24)(10,55,40,25)(11,56,41,26)(12,57,42,27)(13,58,43,28)(14,59,44,29)(15,60,45,30)(61,76,91,106)(62,77,92,107)(63,78,93,108)(64,79,94,109)(65,80,95,110)(66,81,96,111)(67,82,97,112)(68,83,98,113)(69,84,99,114)(70,85,100,115)(71,86,101,116)(72,87,102,117)(73,88,103,118)(74,89,104,119)(75,90,105,120)(121,136,151,166)(122,137,152,167)(123,138,153,168)(124,139,154,169)(125,140,155,170)(126,141,156,171)(127,142,157,172)(128,143,158,173)(129,144,159,174)(130,145,160,175)(131,146,161,176)(132,147,162,177)(133,148,163,178)(134,149,164,179)(135,150,165,180)(181,226,211,196)(182,227,212,197)(183,228,213,198)(184,229,214,199)(185,230,215,200)(186,231,216,201)(187,232,217,202)(188,233,218,203)(189,234,219,204)(190,235,220,205)(191,236,221,206)(192,237,222,207)(193,238,223,208)(194,239,224,209)(195,240,225,210), (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), (1,88,31,118)(2,117,32,87)(3,86,33,116)(4,115,34,85)(5,84,35,114)(6,113,36,83)(7,82,37,112)(8,111,38,81)(9,80,39,110)(10,109,40,79)(11,78,41,108)(12,107,42,77)(13,76,43,106)(14,105,44,75)(15,74,45,104)(16,103,46,73)(17,72,47,102)(18,101,48,71)(19,70,49,100)(20,99,50,69)(21,68,51,98)(22,97,52,67)(23,66,53,96)(24,95,54,65)(25,64,55,94)(26,93,56,63)(27,62,57,92)(28,91,58,61)(29,120,59,90)(30,89,60,119)(121,188,151,218)(122,217,152,187)(123,186,153,216)(124,215,154,185)(125,184,155,214)(126,213,156,183)(127,182,157,212)(128,211,158,181)(129,240,159,210)(130,209,160,239)(131,238,161,208)(132,207,162,237)(133,236,163,206)(134,205,164,235)(135,234,165,204)(136,203,166,233)(137,232,167,202)(138,201,168,231)(139,230,169,200)(140,199,170,229)(141,228,171,198)(142,197,172,227)(143,226,173,196)(144,195,174,225)(145,224,175,194)(146,193,176,223)(147,222,177,192)(148,191,178,221)(149,220,179,190)(150,189,180,219) );
G=PermutationGroup([[(1,231,31,201),(2,202,32,232),(3,233,33,203),(4,204,34,234),(5,235,35,205),(6,206,36,236),(7,237,37,207),(8,208,38,238),(9,239,39,209),(10,210,40,240),(11,181,41,211),(12,212,42,182),(13,183,43,213),(14,214,44,184),(15,185,45,215),(16,216,46,186),(17,187,47,217),(18,218,48,188),(19,189,49,219),(20,220,50,190),(21,191,51,221),(22,222,52,192),(23,193,53,223),(24,224,54,194),(25,195,55,225),(26,226,56,196),(27,197,57,227),(28,228,58,198),(29,199,59,229),(30,230,60,200),(61,171,91,141),(62,142,92,172),(63,173,93,143),(64,144,94,174),(65,175,95,145),(66,146,96,176),(67,177,97,147),(68,148,98,178),(69,179,99,149),(70,150,100,180),(71,121,101,151),(72,152,102,122),(73,123,103,153),(74,154,104,124),(75,125,105,155),(76,156,106,126),(77,127,107,157),(78,158,108,128),(79,129,109,159),(80,160,110,130),(81,131,111,161),(82,162,112,132),(83,133,113,163),(84,164,114,134),(85,135,115,165),(86,166,116,136),(87,137,117,167),(88,168,118,138),(89,139,119,169),(90,170,120,140)], [(1,46,31,16),(2,47,32,17),(3,48,33,18),(4,49,34,19),(5,50,35,20),(6,51,36,21),(7,52,37,22),(8,53,38,23),(9,54,39,24),(10,55,40,25),(11,56,41,26),(12,57,42,27),(13,58,43,28),(14,59,44,29),(15,60,45,30),(61,76,91,106),(62,77,92,107),(63,78,93,108),(64,79,94,109),(65,80,95,110),(66,81,96,111),(67,82,97,112),(68,83,98,113),(69,84,99,114),(70,85,100,115),(71,86,101,116),(72,87,102,117),(73,88,103,118),(74,89,104,119),(75,90,105,120),(121,136,151,166),(122,137,152,167),(123,138,153,168),(124,139,154,169),(125,140,155,170),(126,141,156,171),(127,142,157,172),(128,143,158,173),(129,144,159,174),(130,145,160,175),(131,146,161,176),(132,147,162,177),(133,148,163,178),(134,149,164,179),(135,150,165,180),(181,226,211,196),(182,227,212,197),(183,228,213,198),(184,229,214,199),(185,230,215,200),(186,231,216,201),(187,232,217,202),(188,233,218,203),(189,234,219,204),(190,235,220,205),(191,236,221,206),(192,237,222,207),(193,238,223,208),(194,239,224,209),(195,240,225,210)], [(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)], [(1,88,31,118),(2,117,32,87),(3,86,33,116),(4,115,34,85),(5,84,35,114),(6,113,36,83),(7,82,37,112),(8,111,38,81),(9,80,39,110),(10,109,40,79),(11,78,41,108),(12,107,42,77),(13,76,43,106),(14,105,44,75),(15,74,45,104),(16,103,46,73),(17,72,47,102),(18,101,48,71),(19,70,49,100),(20,99,50,69),(21,68,51,98),(22,97,52,67),(23,66,53,96),(24,95,54,65),(25,64,55,94),(26,93,56,63),(27,62,57,92),(28,91,58,61),(29,120,59,90),(30,89,60,119),(121,188,151,218),(122,217,152,187),(123,186,153,216),(124,215,154,185),(125,184,155,214),(126,213,156,183),(127,182,157,212),(128,211,158,181),(129,240,159,210),(130,209,160,239),(131,238,161,208),(132,207,162,237),(133,236,163,206),(134,205,164,235),(135,234,165,204),(136,203,166,233),(137,232,167,202),(138,201,168,231),(139,230,169,200),(140,199,170,229),(141,228,171,198),(142,197,172,227),(143,226,173,196),(144,195,174,225),(145,224,175,194),(146,193,176,223),(147,222,177,192),(148,191,178,221),(149,220,179,190),(150,189,180,219)]])
87 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 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 | 2 | 3 | 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 | 2 | 30 | 30 | 30 | 30 | 2 | 2 | ··· | 2 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
87 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D10 | D10 | D15 | D30 | D30 | 2- 1+4 | Q8.15D6 | Q8.10D10 | Q8.15D30 |
| kernel | Q8.15D30 | D60⋊11C2 | Q8×D15 | Q8⋊3D15 | Q8×C30 | Q8×C10 | C6×Q8 | C2×C20 | C5×Q8 | C2×C12 | C3×Q8 | C2×Q8 | C2×C4 | Q8 | C15 | C5 | C3 | C1 |
| # reps | 1 | 6 | 4 | 4 | 1 | 1 | 2 | 3 | 4 | 6 | 8 | 4 | 12 | 16 | 1 | 2 | 4 | 8 |
Matrix representation of Q8.15D30 ►in GL4(𝔽61) generated by
| 0 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
| 19 | 59 | 0 | 0 |
| 59 | 42 | 0 | 0 |
| 0 | 0 | 42 | 2 |
| 0 | 0 | 2 | 19 |
| 60 | 29 | 0 | 0 |
| 29 | 1 | 0 | 0 |
| 0 | 0 | 5 | 38 |
| 0 | 0 | 38 | 56 |
| 0 | 0 | 5 | 38 |
| 0 | 0 | 38 | 56 |
| 60 | 29 | 0 | 0 |
| 29 | 1 | 0 | 0 |
G:=sub<GL(4,GF(61))| [0,60,0,0,1,0,0,0,0,0,0,60,0,0,1,0],[19,59,0,0,59,42,0,0,0,0,42,2,0,0,2,19],[60,29,0,0,29,1,0,0,0,0,5,38,0,0,38,56],[0,0,60,29,0,0,29,1,5,38,0,0,38,56,0,0] >;
Q8.15D30 in GAP, Magma, Sage, TeX
Q_8._{15}D_{30} % in TeX
G:=Group("Q8.15D30"); // GroupNames label
G:=SmallGroup(480,1174);
// by ID
G=gap.SmallGroup(480,1174);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,219,100,675,2693,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=1,b^2=c^30=d^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^29>;
// generators/relations